@article{dong_li_ruiz-alvarez_2023, title={A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces}, volume={7}, ISSN={["2661-8893"]}, url={https://doi.org/10.1007/s42967-023-00281-x}, DOI={10.1007/s42967-023-00281-x}, journal={COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION}, author={Dong, Baiying and Li, Zhilin and Ruiz-Alvarez, Juan}, year={2023}, month={Jul} }
 @article{li_li_pan_2023, title={Accurate derivatives approximations and applications to some elliptic PDEs using HOC methods}, volume={459}, ISSN={["1873-5649"]}, url={https://doi.org/10.1016/j.amc.2023.128265}, DOI={10.1016/j.amc.2023.128265}, abstractNote={For many application problems that are modeled by partial differential equations (PDEs), not only it is important to obtain accurate approximations to the solutions, but also accurate approximations to the derivatives of the solutions. In this study, some new high order compact (HOC) finite difference schemes are derived to approximate the first and second derivatives of the solution to some elliptic PDEs using the numerical solution obtained from a HOC scheme applied to the same PDE. Convergence analysis for the computed derivatives is also presented to show that the order of the convergence is the same as that of the solution. The new HOC schemes for computing partial derivatives at both interior and boundary grid points take into account of the partial differential equations including the source term and/or the boundary conditions (Dirichlet, Neumann, or Robin). Fourth order accurate compact finite difference formulas with pre-computed coefficients and weights are developed for Poisson/Helmholtz PDEs, and code generated coefficients for diffusion-advection equations with constant coefficients. One important application is a new fourth-order compact scheme for solving incompressible Stokes equations with periodic boundary conditions.}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Li, Jin and Li, Zhilin and Pan, Kejia}, year={2023}, month={Dec} }
 @article{amat_li_ruiz-alvarez_solano_trillo_2023, title={Adapting Cubic Hermite Splines to the Presence of Singularities Through Correction Terms}, volume={95}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-023-02191-9}, abstractNote={Abstract Hermite interpolation is classically used to reconstruct smooth data when the function and its first order derivatives are available at certain nodes. If first order derivatives are not available, it is easy to set a system of equations imposing some regularity conditions at the data nodes in order to obtain them. This process leads to the construction of a Hermite spline. The problem of the described Hermite splines is that the accuracy is lost if the data contains singularities. The consequence is the appearance of oscillations, if there is a jump discontinuity in the function, that globally affects the accuracy of the spline, or the smearing of singularities, if the discontinuities are in the derivatives of the function. This paper is devoted to the construction and analysis of a new technique that allows for the computation of accurate first order derivatives of a function close to singularities using a Hermite spline. The idea is to correct the system of equations of the spline in order to attain the desired accuracy even close to the singularities. Once we have computed the first order derivatives with enough accuracy, a correction term is added to the Hermite spline in the intervals that contain a singularity. The aim is to reconstruct piecewise smooth functions with $$O(h^4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> accuracy even close to the singularities. The process of adaption will require some knowledge about the position of the singularity and the jumps of the function and some of its derivatives at the singularity. The whole process can be used as a post-processing, where a correction term is added to the classical cubic Hermite spline. Proofs for the accuracy and regularity of the corrected spline and its derivatives are given. We also analyse the mechanism that eliminates the Gibbs phenomenon close to jump discontinuities in the function. The numerical experiments presented confirm the theoretical results obtained.}, number={3}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Amat, Sergio and Li, Zhilin and Ruiz-Alvarez, Juan and Solano, Concepcion and Trillo, Juan C.}, year={2023}, month={Jun} }
 @article{hu_pan_wu_ge_li_2023, title={An efficient extrapolation multigrid method based on a HOC scheme on nonuniform rectilinear grids for solving 3D anisotropic convection-diffusion problems}, volume={403}, ISSN={["1879-2138"]}, DOI={10.1016/j.cma.2022.115724}, abstractNote={We develop an efficient multigrid method combined with a high-order compact (HOC) finite difference scheme on nonuniform rectilinear grids for solving 3D diagonal anisotropic convection–diffusion problems with boundary/interior layers. Firstly, we derive a fourth-order compact finite difference scheme to discretize the 3D anisotropic convection–diffusion equation on a rectilinear grid. Then, the resulting large-scale asymmetric linear system of equations is solved by a generalized extrapolation cascadic multigrid (gEXCMG) method based on two novel multigrid (MG) prolongation operators. The highlight of this paper is the application of the quintic Lagrange interpolation and the completed Richardson extrapolation in the design of the new MG prolongation operator on nonuniform rectilinear grids, which can produce a good initial guess (sixth-order approximation to the finite difference solution) for the SSOR-preconditioned biconjugate gradient stabilized (BiCGStab) smoother. In the end, numerical experiments show that the gEXCMG method combined with the HOC scheme can achieve fourth-order accuracy for 3D anisotropic convection–diffusion problems with few smoothing steps on the finest grid. Moreover, the proposed gEXCMG method can offer substantially better efficiency than the state-of-the-art algebraic MG solver, namely, aggregation-based algebraic multigrid (AGMG) method, for large linear systems arising from the discretization of second order elliptic PDEs.}, journal={COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING}, author={Hu, Shuanggui and Pan, Kejia and Wu, Xiaoxin and Ge, Yongbin and Li, Zhilin}, year={2023}, month={Jan} }
 @article{ji_wang_chen_li_2023, title={Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems}, volume={57}, ISSN={["2804-7214"]}, DOI={10.1051/m2an/2023047}, abstractNote={In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix–Raviart and the rotated- Q 1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H 1 - and L 2 -norms are proved and confirmed with numerical experiments.}, number={4}, journal={ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS}, author={Ji, Haifeng and Wang, Feng and Chen, Jinru and Li, Zhilin}, year={2023}, month={Jul}, pages={2041–2076} }
 @article{li_pan_2023, title={HIGH ORDER COMPACT SCHEMES FOR FLUX TYPE BCS}, volume={45}, ISSN={["1095-7197"]}, url={https://doi.org/10.1137/21M1444771}, DOI={10.1137/21M1444771}, number={2}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Li, Zhilin and Pan, Kejia}, year={2023}, pages={A646–A674} }
 @article{dong_li_ruiz-álvarez_2023, title={Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems}, volume={2}, url={https://ojs.wiserpub.com/index.php/RRCS/article/view/2118}, DOI={10.37256/rrcs.212023}, number={1}, journal={Research Reports on Computer Science}, author={Dong, Baiying and Li, Zhilin and Ruiz-Álvarez, Juan}, year={2023}, month={Jan}, pages={15–27} }
 @article{dong_li_ruiz-álvarez_2023, title={Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems}, url={https://doi.org/10.37256/rrcs.2120232118}, DOI={10.37256/rrcs.2120232118}, abstractNote={In this paper, third-order compact and fourth-order finite element methods (FEMs) based on simple modifications of traditional FEMs are proposed for solving one-dimensional Sturm-Liouville boundary value problems (BVPs). The key idea is based on interpolation error estimates. A simple posterior error analysis of the original piecewise linear finite element space leads to a third-order accurate solution in the L2 norm, second-order in the H1, and the energy norm. The novel idea is also applied to obtain a fourth-order FEM based on the quadratic finite element space. The basis functions in the new fourth-order FEM are more compact compared with that of the classic cubic basis functions. Numerical examples presented in this paper have confirmed the convergence order and analysis. A generalization to a class of nonlinear two-point BVPs is also discussed and tested.}, journal={Research Reports on Computer Science}, author={Dong, Baiying and Li, Zhilin and Ruiz-Álvarez, Juan}, year={2023}, month={Jan} }
 @article{pan_fu_li_hu_li_2023, title={New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations}, volume={16}, ISSN={["2079-7338"]}, DOI={10.4208/nmtma.OA-2022-0073}, number={2}, journal={NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS}, author={Pan, Kejia and Fu, Kang and Li, Jin and Hu, Hongling and Li, Zhilin}, year={2023}, month={May}, pages={393–409} }
 @article{li_mikayelyan_2023, title={Numerical analysis of a free boundary problem with non-local obstacles}, volume={135}, ISSN={["1873-5452"]}, url={https://doi.org/10.1016/j.aml.2022.108414}, DOI={10.1016/j.aml.2022.108414}, abstractNote={The paper deals with the obstacle-like minimization problem in the cylindrical domain Ω=D×(−l,l) J(u)=∫Ω|∇u|2dx+2∫Dmax{v(x′),0}dx′, where x=(x′,xn), and v(x′)=∫−llu(x′,xn)dxn. The corresponding Euler–Lagrange equation is Δu(x′,xn)=χ{v>0}(x′)+−∂xnu(x′,−l)+∂xnu(x′,l)χ{v=0}(x′). Due to the non-local nature of the obstacle, the comparison principle does not hold for the minimizers u(x), which makes the problem challenging both analytically and numerically. The standard optimization techniques such as Newton or quasi-Newton’s methods require approximations of the Jacobians that are four dimensional tensors and are prohibitively expensive both in storage and computational time due to the nature of the three dimensional problem. In this paper, a new algorithm that can compute the global minimum is introduced. Non-trivial exact solutions have been constructed; and second order accuracy has been confirmed. Another important contribution is the numerical testing of the comparison principle for functions v(x′), as conjectured by M. Chipot and the second author in Chipot and Mikayelyan (2022).}, journal={APPLIED MATHEMATICS LETTERS}, author={Li, Zhilin and Mikayelyan, Hayk}, year={2023}, month={Jan} }
 @article{amat_li_ruiz-alvarez_solano_trillo_2023, title={Numerical integration rules with improved accuracy close to discontinuities}, volume={210}, ISSN={["1872-7166"]}, url={https://doi.org/10.1016/j.matcom.2023.03.032}, DOI={10.1016/j.matcom.2023.03.032}, abstractNote={This work is devoted to the construction and analysis of a new nonlinear technique that allows obtaining accurate numerical integrations of any order using data that contains discontinuities, and when the integrand is only known at grid points. The novelty of the technique consists in the inclusion of correction terms with a closed expression that depend on the size of the jumps of the function and its derivatives at the discontinuities, that are supposed to be known. The addition of these terms allows recovering the accuracy of classical numerical integration formulas close to the discontinuities, as these correction terms account for the error that the classical integration formulas commit up to their accuracy at smooth zones. Thus, the correction terms can be added during the integration or as post-processing, which is useful if the main calculation of the integral has been already done using classical formulas. We include several numerical experiments that confirm the theoretical conclusions reached in this article.}, journal={MATHEMATICS AND COMPUTERS IN SIMULATION}, author={Amat, Sergio and Li, Zhilin and Ruiz-Alvarez, Juan and Solano, Concepcion and Trillo, Juan C.}, year={2023}, month={Aug}, pages={593–614} }
 @article{li_pan_ruiz-alvarez_2023, title={Stable high order FD methods for interface and internal layer problems based on non-matching grids}, volume={11}, ISSN={["1572-9265"]}, DOI={10.1007/s11075-023-01680-0}, journal={NUMERICAL ALGORITHMS}, author={Li, Zhilin and Pan, Kejia and Ruiz-Alvarez, Juan}, year={2023}, month={Nov} }
 @article{li_yin_li_2022, title={A Variant Modified Skew-Normal Splitting Iterative Method for Non-Hermitian Positive Definite Linear Systems}, volume={2}, ISSN={["2079-7338"]}, DOI={10.4208/nmtma.OA-2021-0038}, journal={NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS}, author={Li, Rui and Yin, Jun-Feng and Li, Zhi-Lin}, year={2022}, month={Feb} }
 @article{pan_wu_hu_yu_li_2022, title={A new FV scheme and fast cell-centered multigrid solver for 3D anisotropic diffusion equations with discontinuous coefficients}, volume={449}, ISSN={["1090-2716"]}, url={https://doi.org/10.1016/j.jcp.2021.110794}, DOI={10.1016/j.jcp.2021.110794}, abstractNote={In this paper, an efficient cell-centered extrapolation cascadic multigrid (CEXCMG) method is proposed for solving large linear system of equations resulting from finite volume (FV) discretizations of three dimensional (3D) anisotropic diffusion equations with discontinuous coefficients. For cell-centered FV schemes, the values at vertex need to be approximated often by a linear combination of neighboring cell-centered values. In the literature, the weighted coefficients are obtained by solving local linear system of equations which is costly in 3D. One of the novelties of this paper is a new approach for obtaining vertex values by interpolating the cell-centered ones, which avoids solving local linear system of equations even with arbitrary diffusion tensors. Another main novelty of this paper is a new cascadic multigrid solver based on a prolongation operator, the newly developed explicit gradient transfer method, and a splitting extrapolation operator for solving 3D anisotropic diffusion equations with discontinuous coefficients. Numerical experiments are presented to demonstrate the efficiency and robustness of the CEXCMG method in terms of the mesh size and the contrast in the coefficients of the anisotropic diffusion tensor.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, publisher={Elsevier BV}, author={Pan, Kejia and Wu, Xiaoxin and Hu, Hongling and Yu, Yunlong and Li, Zhilin}, year={2022}, month={Jan} }
 @article{ji_wang_chen_li_2022, title={A new parameter free partially penalized immersed finite element and the optimal convergence analysis}, volume={150}, ISSN={["0945-3245"]}, DOI={10.1007/s00211-022-01276-1}, abstractNote={This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. A lifting operator is introduced on interface edges to ensure the coercivity of the method without requiring an ad-hoc stabilization parameter. The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature. A new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements. Optimal error estimates are derived rigorously with the constant independent of the interface location relative to the mesh. The new method and analysis have also been extended to variable coefficients and three-dimensional problems. Numerical examples are also provided to confirm the theoretical analysis and efficiency of the new method.}, number={4}, journal={NUMERISCHE MATHEMATIK}, author={Ji, Haifeng and Wang, Feng and Chen, Jinru and Li, Zhilin}, year={2022}, month={Apr}, pages={1035–1086} }
 @article{singh_singh_li_2022, title={A new patch up technique for elliptic partial differential equation with irregularities}, volume={407}, ISSN={["1879-1778"]}, DOI={10.1016/j.cam.2021.113975}, abstractNote={This paper presents a new technique based on a collocation method using cubic splines for second order elliptic equation with irregularities in one dimension and two dimensions. The differential equation is first collocated at the two smooth sub domains divided by the interface. We extend the sub domains from the interior of the domain and then the scheme at the interface is developed by patching them up. The scheme obtained gives the second order accurate solution at the interface as well as at the regular points. Second order accuracy for the approximations of the first order and the second order derivative of the solution can also be seen from the experiments performed. Numerical experiments for 2D problems also demonstrate the second order accuracy of the present scheme for the solution u and the derivatives ux,uxx and the mixed derivative uxy. The approach to derive the interface relations, established in this paper for elliptic interface problems, can be helpful to derive high order accurate numerical methods. Numerical tests exhibit the super convergent properties of the scheme.}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Singh, Swarn and Singh, Suruchi and Li, Zhilin}, year={2022}, month={Jun} }
 @article{dong_feng_li_2022, title={AN L-8 SECOND ORDER CARTESIAN METHOD FOR 3D ANISOTROPIC INTERFACE PROBLEMS}, volume={40}, ISSN={["1991-7139"]}, DOI={10.4208/jcm.2103-m2020-0107}, number={6}, journal={JOURNAL OF COMPUTATIONAL MATHEMATICS}, author={Dong, Baiying and Feng, Xiufeng and Li, Zhilin}, year={2022}, pages={882–912} }
 @article{ji_wang_chen_li_2022, title={An immersed C R-P-0 element for Stokes interface problems and the optimal convergence analysis}, volume={399}, ISSN={["1879-2138"]}, DOI={10.1016/j.cma.2022.115306}, abstractNote={This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional CR−P0 element with modifications near the interface according to the interface jump conditions. We prove that the IFE basis functions are unisolvent on arbitrary triangles without any angle conditions and the IFE spaces have the optimal approximation capabilities, although the proof is challenging due to the coupling of the velocity and the pressure. The stability and the optimal error estimates of the proposed IFE method are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.}, journal={COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING}, author={Ji, Haifeng and Wang, Feng and Chen, Jinru and Li, Zhilin}, year={2022}, month={Sep} }
 @article{pan_he_li_2021, title={A High Order Compact FD Framework for Elliptic BVPs Involving Singular Sources, Interfaces, and Irregular Domains}, volume={88}, ISSN={["1573-7691"]}, url={https://doi.org/10.1007/s10915-021-01570-4}, DOI={10.1007/s10915-021-01570-4}, number={3}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, publisher={Springer Science and Business Media LLC}, author={Pan, Kejia and He, Dongdong and Li, Zhilin}, year={2021}, month={Sep} }
 @article{li_li_yin_2021, title={A generalized modulus-based Newton method for solving a class of non-linear complementarity problems with P-matrices}, volume={6}, ISSN={["1572-9265"]}, DOI={10.1007/s11075-021-01136-3}, journal={NUMERICAL ALGORITHMS}, author={Li, Rui and Li, Zhi-Lin and Yin, Jun-Feng}, year={2021}, month={Jun} }
 @article{zhang_li_yue_2021, title={Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems}, volume={14}, ISSN={["2079-7338"]}, DOI={10.4208/nmtma.OA-2020-0112}, number={3}, journal={NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS}, author={Zhang, Changjuan and Li, Zhilin and Yue, Xingye}, year={2021}, month={Aug}, pages={773–796} }
 @article{deng_li_pan_2021, title={An ADI-Yee's scheme for Maxwell's equations with discontinuous coefficients}, volume={438}, ISSN={["1090-2716"]}, url={https://doi.org/10.1016/j.jcp.2021.110356}, DOI={10.1016/j.jcp.2021.110356}, abstractNote={An alternating directional implicit (ADI)-Yee's scheme is developed for Maxwell's equations with discontinuous material coefficients along one or several interfaces. In order to use Yee's scheme with the presence of discontinuities, some intermediate quantities along the interface are introduced. The intermediate quantities are from the solutions and their derivatives on the interface and should satisfy some interface conditions. In discretization, those quantities are actually determined implicitly. For a fixed interface and a fixed time step size, the linear system of equations for the intermediate quantities can be pre-determined, so is the PLU or SVD decomposition of the coefficient matrix of the linear system. The ADI-Yee's scheme maintains the structure (the finite difference scheme with modified right-hand sides) as well as the accuracy and stability of Yee's scheme even with the presence of discontinuities. Theoretical analysis and numerical examples are also provided.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, publisher={Elsevier BV}, author={Deng, Shaozhong and Li, Zhilin and Pan, Kejia}, year={2021}, month={Aug} }
 @book{li_norris_2021, title={An Introduction to Partial Differential Equations (with Maple)}, ISBN={9789811228629 9789811228636}, url={http://dx.doi.org/10.1142/12052}, DOI={10.1142/12052}, publisher={WORLD SCIENTIFIC}, author={Li, Zhilin and Norris, Larry}, year={2021}, month={Sep} }
 @article{tong_feng_li_2021, title={Fourth order compact FD methods for convection diffusion equations with variable coefficients}, volume={121}, ISSN={["1873-5452"]}, url={https://doi.org/10.1016/j.aml.2021.107413}, DOI={10.1016/j.aml.2021.107413}, abstractNote={Fourth order finite difference methods combined with an integrating factor strategy for steady convection and diffusion partial differential equations with variable coefficients in both 2D and 3D are proposed using uniform Cartesian grids. An integrating factor strategy is applied to transform the convection and diffusion PDE to a self-adjoint form. Then, a fourth order finite difference method is obtained through a second order scheme followed by the Richardson extrapolation. Another approach is a direct fourth order compact finite difference scheme. The developed integrating factor strategy provides an efficient way for dealing with large convection coefficients. Several numerical examples are presented to demonstrate the convergence order and compare the two fourth order methods.}, journal={APPLIED MATHEMATICS LETTERS}, publisher={Elsevier BV}, author={Tong, Fenghua and Feng, Xinlong and Li, Zhilin}, year={2021}, month={Nov} }
 @article{jiang_yang_li_2021, title={Non-parallel hyperplanes ordinal regression machine}, volume={216}, ISSN={["1872-7409"]}, url={https://doi.org/10.1016/j.knosys.2020.106593}, DOI={10.1016/j.knosys.2020.106593}, abstractNote={This paper proposes a method to solve ordinal regression problems, namely the non-parallel hyperplanes ordinal regression machine (NPHORM). The goal of this approach is to find K different hyperplanes for the K classes with ordinal information, so that each class is as close as possible to the corresponding hyperplane while as far as possible from the adjacent to the left and right classes. The more flexible separate hyperplanes are preferred using the order information of the data. As a result, this approach only needs to solve K quadratic programming problems independently. Our approach NPHORM is validated on 2 artificial datasets, 16 discretized regression datasets and 17 real ordinal regression datasets and compared with 8 outstanding SVM-based ordinal regression approaches. The results show that our approach NPHORM is comparable with the other SVM-based approaches, especially in real ordinal regression datasets. In addition, our NPHORM is also carried out on the historical color image dataset to compare the performance of deep learning method. Experimental results demonstrate that the performance of our NPHORM outperforms the deep learning methods on MAE.}, journal={KNOWLEDGE-BASED SYSTEMS}, publisher={Elsevier BV}, author={Jiang, Haitao and Yang, Zhixia and Li, Zhilin}, year={2021}, month={Mar} }
 @article{xu_su_li_2021, title={Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D/3D MHD equations}, volume={11}, ISSN={["1572-9265"]}, DOI={10.1007/s11075-021-01224-4}, journal={NUMERICAL ALGORITHMS}, author={Xu, Jiali and Su, Haiyan and Li, Zhilin}, year={2021}, month={Nov} }
 @article{huang_li_2021, title={Partially penalized IFE methods and convergence analysis for elasticity interface problems}, volume={382}, ISSN={["1879-1778"]}, DOI={10.1016/j.cam.2020.113059}, abstractNote={In this paper, some partial penalized immersed finite element methods (PPIFEMs) are proposed and analyzed for solving elasticity interface problems. Through verifying the inverse trace inequality on the interface edges, the optimal convergence in the energy norm is derived. A new test function is constructed to obtain the discrete inf–sup condition of the penalty-free nonsymmetric PPIFEM and is utilized in the proof of the optimal convergence. Furthermore, the effect of the Lamé parameters on convergence is also studied. Various numerical examples and comparisons are provided to confirm the theoretical results.}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Huang, Peiqi and Li, Zhilin}, year={2021}, month={Jan} }
 @article{partially penalized ife methods and convergence analysis for elasticity interface problems_2021, journal={Journal of Computational and Applied Mathematics}, year={2021}, month={Jan} }
 @article{ye_yang_li_2021, title={Quadratic hyper-surface kernel-free least squares support vector regression}, volume={25}, ISSN={["1571-4128"]}, DOI={10.3233/IDA-205094}, abstractNote={We present a novel kernel-free regressor, called quadratic hyper-surface kernel-free least squares support vector regression (QLSSVR), for some regression problems. The task of this approach is to find a quadratic function as the regression function, which is obtained by solving a quadratic programming problem with the equality constraints. Basically, the new model just needs to solve a system of linear equations to achieve the optimal solution instead of solving a quadratic programming problem. Therefore, compared with the standard support vector regression, our approach is much efficient due to kernel-free and solving a set of linear equations. Numerical results illustrate that our approach has better performance than other existing regression approaches in terms of regression criterion and CPU time.}, number={2}, journal={INTELLIGENT DATA ANALYSIS}, author={Ye, Junyou and Yang, Zhixia and Li, Zhilin}, year={2021}, pages={265–281} }
 @article{xiao_feng_li_2021, title={The local tangential lifting method for moving interface problems on surfaces with applications}, volume={431}, ISSN={["1090-2716"]}, url={https://doi.org/10.1016/j.jcp.2021.110146}, DOI={10.1016/j.jcp.2021.110146}, abstractNote={Abstract In this paper, a new numerical computational frame is presented for solving moving interface problems modeled by parabolic PDEs on static and evolving surfaces. The surface PDEs can have Dirac delta source distributions and discontinuous coefficients. One application is for thermally driven moving interfaces on surfaces such as Stefan problems and dendritic solidification phenomena on solid surfaces. One novelty of the new method is the local tangential lifting method to construct discrete delta functions on surfaces. The idea of the local tangential lifting method is to transform a local surface problem to a local two dimensional one on the tangent planes of surfaces at some selected surface nodes. Moreover, a surface version of the front tracking method is developed to track moving interfaces on surfaces. Strategies have been developed for computing geodesic curvatures of interfaces on surfaces. Various numerical examples are presented to demonstrate the accuracy of the new methods. It is also interesting to see the comparison of the dendritic solidification processes in two dimensional spaces and on surfaces.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, publisher={Elsevier BV}, author={Xiao, Xufeng and Feng, Xinlong and Li, Zhilin}, year={2021}, month={Apr} }
 @article{dong_feng_li_2020, title={AN FE-FD METHOD FOR ANISOTROPIC ELLIPTIC INTERFACE PROBLEMS}, volume={42}, ISSN={["1095-7197"]}, DOI={10.1137/19M1291030}, abstractNote={Anisotropic elliptic interface problems are important but hard to solve either analytically or numerically. There is limited literature on numerical methods based on structured meshes. Finite element methods are often used, but the usual average error estimates cannot guarantee accuracy of the solution near or at the interface. For finite difference methods, it is challenging to discretize mixed derivatives and carry out the convergence analysis. In this paper, a new finite element-finite difference (FE-FD) method that combines a finite element discretization (away from the interface) whose coefficient matrix is a symmetric semipositive definite, with a finite difference discretization (near or on the interface) whose coefficient matrix part has properties of an M-matrix, is developed. An interpolation scheme based on the immersed interface method is also applied to compute the normal derivative of solution (or gradient) accurately from each side of the interface. Error analysis and numerical experiments are also presented.}, number={4}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Dong, Baiying and Feng, Xiufang and Li, Zhilin}, year={2020}, pages={B1041–B1066} }
 @article{tong_wang_feng_zhao_li_2020, title={How to obtain an accurate gradient for interface problems?}, volume={405}, url={https://doi.org/10.1016/j.jcp.2019.109070}, DOI={10.1016/j.jcp.2019.109070}, abstractNote={It is well-known that the Immersed Interface Method (IIM) is second order accurate for interface problems. But the accuracy of the first order derivatives, or gradients for short, is not so clear and is often assumed to be first order accurate. In this paper, new strategies based on IIM are proposed for elliptic interface problems to compute the gradient at grid points both regular and irregular, and at the interface from each side of the interface. Second order in 1D, or nearly second order (except a factor of |log⁡h|) convergence in 2D of the computed gradient is obtained with almost no extra cost, and has been explained in intuition and verified by non-trivial numerical tests. Numerical examples in one, two dimensions, radial and axis-symmetric cases in polar and spherical coordinates are presented to validate the numerical methods and analysis.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Tong, Fenghua and Wang, Weilong and Feng, Xinlong and Zhao, Jianping and Li, Zhilin}, year={2020}, month={Mar}, pages={109070} }
 @article{chen_feng_li_2019, title={A direct method for accurate solution and gradient computations for elliptic interface problems}, volume={80}, ISSN={["1572-9265"]}, DOI={10.1007/s11075-018-0503-5}, number={3}, journal={NUMERICAL ALGORITHMS}, author={Chen, Xiaohong and Feng, Xiufang and Li, Zhilin}, year={2019}, month={Mar}, pages={709–740} }
 @article{xiao_feng_li_2019, title={A gradient recovery-based adaptive finite element method for convection-diffusion-reaction equations on surfaces}, volume={120}, ISSN={["1097-0207"]}, DOI={10.1002/nme.6163}, abstractNote={Summary In this paper, we present an adaptive mesh refinement method for solving convection‐diffusion‐reaction equations on surfaces, which is a fundamental subproblem in many models for simulating the transport of substances on biological films and solid surfaces. The method considered is a combination of well‐known techniques: the surface finite element method, streamline diffusion stabilization, and the gradient recovery–based Zienkiewicz‐Zhu error estimator. The streamline diffusion method overcomes the instability issue of the finite element method for the dominance of the convection. The gradient recovery–based adaptive mesh refinement strategy enables the method to provide high‐resolution numerical solutions by relatively fewer degrees of freedom. Moreover, the implementation detail of a surface mesh refinement technique is presented. Various numerical examples, including the convection‐dominated diffusion problems with large variations of solutions, nearly singular solutions, discontinuous sources, and internal layers on surfaces, are presented to demonstrate the efficacy and accuracy of the proposed method.}, number={7}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING}, author={Xiao, Xufeng and Feng, Xinlong and Li, Zhilin}, year={2019}, month={Nov}, pages={901–917} }
 @article{li_dong_tong_wang_2019, title={An Augmented IB Method & Analysis for Elliptic BVP on Irregular Domains}, volume={119}, ISSN={["1526-1506"]}, DOI={10.32604/cmes.2019.04635}, abstractNote={The immersed boundary method is well-known, popular, and has had vast areas of applications due to its simplicity and robustness even though it is only first order accurate near the interface. In this paper, an immersed boundary-augmented method has been developed for linear elliptic boundary value problems on arbitrary domains (exterior or interior) with a Dirichlet boundary condition. The new method inherits the simplicity, robustness, and first order convergence of the IB method but also provides asymptotic first order convergence of partial derivatives. Numerical examples are provided to confirm the analysis.}, number={1}, journal={CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES}, author={Li, Zhilin and Dong, Baiying and Tong, Fenghua and Wang, Weilong}, year={2019}, pages={63–72} }
 @article{yao_zhang_tian_zhou_li_wang_2019, title={Analysis of Network Structure of Urban Bike-Sharing System: A Case Study Based on Real-Time Data of a Public Bicycle System}, volume={11}, ISSN={["2071-1050"]}, DOI={10.3390/su11195425}, abstractNote={To better understand the characteristics of a bike-sharing system, we applied complex network methods to analyze the relationship between stations within the bike-sharing system. Firstly, using Gephi software, we constructed the public bicycle networks of different urban areas based on the real-time data of the Nanjing public bicycle system. Secondly, we analyzed and compared degree, strength, radiation distance, and community structure of the networks to understand the internal relations of the public bicycle system. The results showed that there were many stations with low usage of public bicycles. Furthermore, there was a geographical division between high-demand and low-demand areas for public bicycles. The usage of public bicycles at a station was not only related to land use but also related to the usage of bicycles at stations nearby. Moreover, the average service coverage of the public bicycle system was consistent with the original intention of “the first and last mile”, and public bicycles could meet different travel needs.}, number={19}, journal={SUSTAINABILITY}, author={Yao, Yi and Zhang, Yifang and Tian, Lixin and Zhou, Nianxing and Li, Zhilin and Wang, Minggang}, year={2019}, month={Oct} }
 @article{zhao_zhang_li_zhang_2019, title={Numerical Validations of the Tangent Linear Model for the Lorenz Equations}, volume={120}, ISSN={["1526-1506"]}, DOI={10.32604/cmes.2019.04483}, number={1}, journal={CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES}, author={Zhao, Tengjin and Zhang, Jing and Li, Zhilin and Zhang, Zhiyue}, year={2019}, pages={83–104} }
 @article{yao_jiang_li_2019, title={Spatiotemporal characteristics of green travel: A classification study on a public bicycle system}, volume={238}, ISSN={["1879-1786"]}, DOI={10.1016/j.jclepro.2019.117892}, abstractNote={Abstract Understanding the characteristics of users and stations provides the foundation for a more efficient public bicycle system. Based on the real-time data of the Nanjing public bicycle system, we presented the spatiotemporal characteristics of users and stations combining data mining and geographic visualization. First, we analyzed users' gender, age, weekly flow, and time-segment flow, and classified the users into different types. In addition, we studied the cycling chains of certain users in details to understand the differences. Second, we analyzed the station distribution, station flow, station time-segment flow, and the surrounding environment, and studied the specific stations of different types to reveal the diverse characteristics. Moreover, we also explored the relationship between the user types and the station types. The results showed that public bicycles were mainly used for commuting or transferring, and social and economic activities around stations greatly influenced the use of public bicycles. However, the usage of the public bicycle system was still at a low level. Furthermore, different types of users had different cycling purposes, and different types of stations showed different characteristics of renting flow and returning flow. At last, we proposed different incentives and management measures for different types of users and stations.}, journal={JOURNAL OF CLEANER PRODUCTION}, author={Yao, Yi and Jiang, Xin and Li, Zhilin}, year={2019}, month={Nov} }
 @article{chen_li_ruiz alvarez_2018, title={A direct IIM approach for two-phase Stokes equations with discontinuous viscosity on staggered grids}, volume={172}, ISSN={["1879-0747"]}, url={https://doi.org/10.1016/j.compfluid.2018.03.038}, DOI={10.1016/j.compfluid.2018.03.038}, abstractNote={In this paper, a direct immersed interface method (IIM) is proposed to solve two-phase incompressible Stokes equations with an interface and a piecewise constant viscosity on staggered grids. The velocity components and the pressure are placed in different grid points and the Marker and Cell (MAC) scheme is used for discretizing the momentum and continuity equations at regular grid points. At irregular grid points, correction terms are added to the finite difference scheme to offset the discontinuities. The correction terms are determined directly by an interpolation scheme using the values of both the velocity and pressure at nearby grid points. The resulted linear system of equations is rank-one deficient and is solved by the Uzawa iterative method. In each Uzawa iteration, an inner GMRES solver is used and preconditioned by the discrete Laplacian. The computed numerical solutions are second order accurate in the L∞ norm for both the velocity and pressure, which is demonstrated in numerical tests. Compared with the augmented interface method (AIIM), one of advantages of this approach is that it avoids the costs for introducing augmented variables and difficulties in solving them from the corresponding Schur complement system. Hence, this new method is easier to implement and computationally more efficient.}, journal={COMPUTERS & FLUIDS}, publisher={Elsevier BV}, author={Chen, Xiaohong and Li, Zhilin and Ruiz Alvarez, Juan}, year={2018}, month={Aug}, pages={549–563} }
 @article{ji_chen_li_2018, title={A high-order source removal finite element method for a class of elliptic interface problems}, volume={130}, ISSN={["1873-5460"]}, DOI={10.1016/j.apnum.2018.03.017}, abstractNote={A high-order finite element method based on unfitted meshes for solving a class of elliptic interface problems whose solution and its normal derivative have finite jumps across an interface is proposed in this paper. The idea of the method is based on the source removal technique first introduced in the immersed interface method (IIM). The strategy is to use the level set representation of the interface and extend the jump conditions that are defined along the interface to a neighborhood of the interface. In our numerical method, the jump conditions only need to be extended to the Lagrange points of elements intersecting with the interface. Optimal error estimates of the method in the broken H1 and L2 norms are rigorously proven. Numerical examples presented in this paper also confirm our theoretical analysis.}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Ji, Haifeng and Chen, Jinru and Li, Zhilin}, year={2018}, month={Aug}, pages={112–130} }
 @article{li_lai_peng_zhang_2018, title={A least squares augmented immersed interface method for solving Navier-Stokes and Darcy coupling equations}, volume={167}, ISSN={["1879-0747"]}, url={https://doi.org/10.1016/j.compfluid.2018.03.032}, DOI={10.1016/j.compfluid.2018.03.032}, abstractNote={A new finite difference method based on Cartesian meshes and fast Poisson/Helmholtz solvers is proposed to solve the coupling of a fluid flow modeled by the incompressible Navier–Stokes equations and a porous media modeled by the Darcy’s law. The finite difference discretization in time is based on the pressure Poisson equation formulation. At each time step, several augmented variables along the interface between the fluid flow and the porous media are introduced so that the coupled equations can be decoupled into several Poisson/Helmholtz equations with those augmented variables acting as jumps of the unknown solution and some directional derivatives. The augmented variables should be chosen so that the Beavers–Joseph–Saffman (BJS) or Beavers–Joseph (BJ) and other interface conditions are satisfied. It has been tested that a direct extension of the augmented idea in [27] does not work well when the fluid flow is modeled by the Navier–Stokes equations. One of the new ideas of this paper is to enforce the divergence free condition at the interface from the fluid side. In this way, the Schur complement matrix for the augmented variables is over-determined and the least squares solution is used for the coupling problem. The new augmented approach enables us to solve the Navier–Stokes and Darcy coupling efficiently with second order accurate velocity and pressure in the L∞ norm for tested problems. The proposed new idea in enforcing the divergence free condition at the interface from the fluid side has also been utilized to solve the Stokes and Darcy coupling equations and shown to outperform the original method in [27]. In additional to the detailed accuracy check for the present method, some interesting numerical simulations for Navier–Stokes and Darcy coupling have been conducted in this paper as well.}, journal={COMPUTERS & FLUIDS}, publisher={Elsevier BV}, author={Li, Zhilin and Lai, Ming-Chih and Peng, Xiaofei and Zhang, Zhiyue}, year={2018}, month={May}, pages={384–399} }
 @article{hu_li_2018, title={Error analysis of the immersed interface method for Stokes equations with an interface}, volume={83}, ISSN={["0893-9659"]}, url={https://doi.org/10.1016/j.aml.2018.03.034}, DOI={10.1016/j.aml.2018.03.034}, abstractNote={The immersed interface method using the three Poisson equation approach has been successfully developed to solve incompressible Stokes equations with interfaces (Li, 2015 and Li and Ito, 2006). While the numerical results show second order convergence for both velocity and pressure, rigorous error analysis is still missing. Based on recent theoretical development, particularly the error analysis by Beale and Layton (2006), second order convergence has been shown in this paper for both pressure and velocity under some assumptions.}, journal={APPLIED MATHEMATICS LETTERS}, publisher={Elsevier BV}, author={Hu, Rui and Li, Zhilin}, year={2018}, month={Sep}, pages={207–211} }
 @book{li_zhonghua_tang_2018, title={Numerical solution of differential equations: Introduction to finite difference and finite element methods}, publisher={New York: Cambridge University Press}, author={Li, Z. and Zhonghua, Q. and Tang, T.}, year={2018} }
 @article{li_chen_zhang_2018, title={ON MULTISCALE ADI METHODS FOR PARABOLIC PDEs WITH A DISCONTINUOUS COEFFICIENT}, volume={16}, ISSN={["1540-3467"]}, DOI={10.1137/17M1151985}, abstractNote={Alternating direction implicit (ADI) method is one of the most efficient methods in solving parabolic PDEs of initial and boundary value problems. However, it is challenging to develop efficient AD...}, number={4}, journal={MULTISCALE MODELING & SIMULATION}, author={Li, Zhilin and Chen, Xiaohong and Zhang, Zhengru}, year={2018}, pages={1623–1647} }
 @article{qin_chen_li_cai_2017, title={A Cartesian grid nonconforming immersed finite element method for planar elasticity interface problems}, volume={73}, ISSN={["1873-7668"]}, DOI={10.1016/j.camwa.2016.11.033}, abstractNote={In this paper, a new nonconforming immersed finite element (IFE) method on triangular Cartesian meshes is developed for solving planar elasticity interface problems. The proposed IFE method possesses optimal approximation property for both compressible and nearly incompressible problems. Its degree of freedom is much less than those of existing finite element methods for the same problem. Moreover, the method is robust with respect to the shape of the interface and its location relative to the domain and the underlying mesh. Both theory and numerical experiments are presented to demonstrate the effectiveness of the new method. Theoretically, the unisolvent property and the consistency of the IFE space are proved. Experimentally, extensive numerical examples are given to show that the approximation orders in L2 norm and semi-H1 norm are optimal under various Lamé parameters settings and different interface geometry configurations.}, number={3}, journal={COMPUTERS & MATHEMATICS WITH APPLICATIONS}, author={Qin, Fangfang and Chen, Jinru and Li, Zhilin and Cai, Mingchao}, year={2017}, month={Feb}, pages={404–418} }
 @article{huang_li_2017, title={A Uniformly Stable Nonconforming FEM Based on Weighted Interior Penalties for Darcy-Stokes-Brinkman Equations}, volume={10}, ISSN={["2079-7338"]}, DOI={10.4208/nmtma.2017.m1610}, abstractNote={Abstract A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.}, number={1}, journal={NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS}, author={Huang, Peiqi and Li, Zhilin}, year={2017}, month={Feb}, pages={22–43} }
 @article{qin_wang_ma_li_2017, title={ACCURATE GRADIENT COMPUTATIONS AT INTERFACES USING FINITE ELEMENT METHODS}, volume={27}, ISSN={["2083-8492"]}, DOI={10.1515/amcs-2017-0037}, abstractNote={Abstract New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution, but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea of Wheeler (1974). For 2D interface problems, the point is to introduce a small tube near the interface and propose the gradient as part of unknowns, which is similar to a mixed finite element method, but only at the interface. Thus the computational cost is just slightly higher than in the standard finite element method. We present a rigorous one dimensional analysis, which shows a second order convergence order for both the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.}, number={3}, journal={INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND COMPUTER SCIENCE}, author={Qin, Fangfang and Wang, Zhaohui and Ma, Zhijie and Li, Zhilin}, year={2017}, month={Sep}, pages={527–537} }
 @article{li_ji_chen_2017, title={ACCURATE SOLUTION AND GRADIENT COMPUTATION FOR ELLIPTIC INTERFACE PROBLEMS WITH VARIABLE COEFFICIENTS}, volume={55}, ISSN={["1095-7170"]}, DOI={10.1137/15m1040244}, abstractNote={A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is to get not only a second order accurate solution but also a second order accurate gradient from each side of the interface. Key to the new method is introducing the jump in the normal derivative of the solution as an augmented variable and rewriting the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivative jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points on or near the interface so that the resulting coefficient matrix is an M-matrix. A multigrid solver is used to solve the linear system of equations, and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface is proved in this paper. Numerical examples for general elliptic interface problems confirm the theoretical analysis and efficiency of the new method.}, number={2}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Li, Zhilin and Ji, Haifeng and Chen, Xiaohong}, year={2017}, pages={570–597} }
 @article{li_qin_2017, title={An Augmented Method for 4th Order PDEs with Discontinuous Coefficients}, volume={73}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-017-0487-7}, number={2-3}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Li, Zhilin and Qin, Fangfang}, year={2017}, month={Dec}, pages={968–979} }
 @article{angot_li_2017, title={An augmented IIM & preconditioning technique for jump embedded boundary conditions}, volume={14}, number={4-5}, journal={International Journal of Numerical Analysis and Modeling}, author={Angot, P. and Li, Z. L.}, year={2017}, pages={712–729} }
 @article{yan_lai_li_zhang_2017, title={New Conservative Finite Volume Element Schemes for the Modified Regularized Long Wave Equation}, volume={9}, ISSN={["2075-1354"]}, DOI={10.4208/aamm.2014.m888}, abstractNote={Abstract In this paper, we propose a new energy-preserving scheme and a new momentum-preserving scheme for the modified regularized long wave equation. The proposed schemes are designed by using the discrete variational derivative method and the finite volume element method. For comparison, we also propose a finite volume element scheme. The conservation properties of the proposed schemes are analyzed and we find that the energy-preserving scheme can precisely conserve the discrete total mass and total energy, the momentum-preserving scheme can precisely conserve the discrete total mass and total momentum, while the finite volume element scheme merely conserve the discrete total mass. We also analyze their linear stability property using the Von Neumann theory and find that the proposed schemes are unconditionally linear stable. Finally, we present some numerical examples to illustrate the effectiveness of the proposed schemes.}, number={2}, journal={ADVANCES IN APPLIED MATHEMATICS AND MECHANICS}, author={Yan, Jinliang and Lai, Ming-Chih and Li, Zhilin and Zhang, Zhiyue}, year={2017}, month={Apr}, pages={250–271} }
 @book{li_qiao_tang_2017, title={Numerical Solution of Differential Equations}, url={http://dx.doi.org/10.1017/9781316678725}, DOI={10.1017/9781316678725}, abstractNote={This introduction to finite difference and finite element methods is aimed at graduate students who need to solve differential equations. The prerequisites are few (basic calculus, linear algebra, and ODEs) and so the book will be accessible and useful to readers from a range of disciplines across science and engineering. Part I begins with finite difference methods. Finite element methods are then introduced in Part II. In each part, the authors begin with a comprehensive discussion of one-dimensional problems, before proceeding to consider two or higher dimensions. An emphasis is placed on numerical algorithms, related mathematical theory, and essential details in the implementation, while some useful packages are also introduced. The authors also provide well-tested MATLAB® codes, all available online.}, journal={Cambridge University Press}, publisher={Cambridge University Press}, author={Li, Zhilin and Qiao, Zhonghua and Tang, Tao}, year={2017}, month={Nov} }
 @article{amat_li_ruiz_2017, title={On an New Algorithm for Function Approximation with Full Accuracy in the Presence of Discontinuities Based on the Immersed Interface Method}, volume={75}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/S10915-017-0596-3}, DOI={10.1007/S10915-017-0596-3}, number={3}, journal={Journal of Scientific Computing}, publisher={Springer Nature}, author={Amat, Sergio and Li, Zhilin and Ruiz, Juan}, year={2017}, month={Nov}, pages={1500–1534} }
 @article{zhang_li_zhang_2016, title={A Sparse Grid Stochastic Collocation Method for Elliptic Interface Problems with Random Input}, volume={67}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-015-0080-x}, number={1}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Zhang, Qian and Li, Zhilin and Zhang, Zhiyue}, year={2016}, month={Apr}, pages={262–280} }
 @article{ji_chen_li_2016, title={A new augmented immersed finite element method without using SVD interpolations}, volume={71}, ISSN={["1572-9265"]}, DOI={10.1007/s11075-015-9999-0}, number={2}, journal={NUMERICAL ALGORITHMS}, author={Ji, Haifeng and Chen, Jinru and Li, Zhilin}, year={2016}, month={Feb}, pages={395–416} }
 @article{li_2016, title={An augmented Cartesian grid method for Stokes-Darcy fluid-structure interactions}, volume={106}, ISSN={["1097-0207"]}, DOI={10.1002/nme.5131}, abstractNote={Summary A new finite difference method based on Cartesian meshes is proposed for solving the fluid–structure interaction between a fluid flow modeled by the Stokes equations and a porous media modeled by the Darcy's law. The idea is to introduce several augmented variables along the interface between the fluid flow and the porous media so that the problem can be decoupled as several Poisson equations. The augmented variables should be chosen so that the Beavers–Joseph–Saffman and other interface conditions are satisfied. In the discretization, the augmented variables have co‐dimension one compared with that of the primitive variables and are solved through the Schur complement system. A non‐trivial analytic solution with a circular interface is constructed to check second‐order convergency of the proposed method. Numerical examples with various interfaces and parameters are also presented. Some simulations show interesting behaviors of the fluid–structure interaction between the fluid flow and the porous media. The computational framework can be applied to other multi‐phase and multi‐physics problems. Copyright © 2015 John Wiley & Sons, Ltd.}, number={7}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING}, author={Li, Zhilin}, year={2016}, month={May}, pages={556–575} }
 @article{zhang_li_2016, title={An augmented iim for helmholtz/poisson equations on irregular domains in complex space}, volume={13}, number={1}, journal={International Journal of Numerical Analysis and Modeling}, author={Zhang, S. D. M. and Li, Z. L.}, year={2016}, pages={166–178} }
 @article{ji_chen_li_2016, title={Augmented immersed finite element methods for some elliptic partial differential equations}, volume={93}, ISSN={["1029-0265"]}, DOI={10.1080/00207160.2015.1005010}, abstractNote={Augmented immersed finite element methods are proposed to solve elliptic interface problems with non-homogeneous jump conditions. The non-homogeneous jump conditions are treated as source terms using the singularity removal technique. For the piecewise constant coefficient case, we transform the original interface problem to a Poisson equation with the same jump in the solution, but an unknown flux jump (augmented variable) which is chosen such that the original flux jump condition is satisfied. The GMRES iterative method is used to solve the augmented variable. The core of each iteration involves solving a Poisson equation using a fast Poisson solver and an interpolation scheme to interpolate the flux jump condition. With a little modification, the method can be applied to solve Poisson equations on irregular domains. Numerical experiments show that not only the computed solution but also the normal derivative are second-order accurate in the L∞ norm.}, number={3}, journal={INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS}, author={Ji, Haifeng and Chen, Jinru and Li, Zhilin}, year={2016}, month={Mar}, pages={540–558} }
 @article{li_mikayelyan_2016, title={Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer}, volume={18}, ISSN={["1463-9971"]}, DOI={10.4171/ifb/357}, abstractNote={A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mapping $w=\sqrt{z}$ in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.}, number={1}, journal={INTERFACES AND FREE BOUNDARIES}, author={Li, Zhilin and Mikayelyan, Hayk}, year={2016}, pages={75–90} }
 @article{su_feng_li_2016, title={Fourth-order compact schemes for Helmholtz equations with piecewise wave numbers in the polar coordinates}, volume={34}, number={5}, journal={Journal of Computational Mathematics}, author={Su, X. L. and Feng, X. F. and Li, Z. L.}, year={2016}, pages={499–510} }
 @article{melnyk_wang_li_xue_2016, title={Prioritization of pesticides based on daily dietary exposure potential as determined from the SHEDS model}, volume={96}, ISSN={["1873-6351"]}, DOI={10.1016/j.fct.2016.07.025}, abstractNote={A major pathway for exposure to many pesticides is through diet. The objectives were to rank pesticides by comparing their calculated daily dietary exposure as determined by EPA's Stochastic Human Exposure and Dose Simulation (SHEDS) to single pesticides for different age groups to acceptable daily intakes (ADI), characterize pesticide trends in exposures over different time periods, and determine commodities contributing to pesticide exposures. SHEDS was applied, using Pesticide Data Program (PDP) (1991-2011) and pesticide usage data on crops from USDA combined with NHANES dietary consumption data, to generate exposure estimates by age group. ADI data collected from EPA, WHO, and other sources were used to rank pesticides based on relativeness of the dietary exposure potential to ADI by age groups. Sensitivity analysis provided trends in pesticide exposures. Within SHEDS, commodities contributing the majority of pesticides with greatest exposure potential were determined. The results indicated that the highest ranking pesticides were methamidophos and diazinon which exceeded 100% of the ADI. Sensitivity analysis indicated that exposure to methamidophos, diazinon, malathion, ethion and formetanate hydrochloride had a marked decrease from 1991-1999 to 2000-2011. Contributions analysis indicated that apples, mushroom, carrots, and lettuce contributed to diazinon exposure. Beans and pepper contributed to methamidophos exposure.}, journal={FOOD AND CHEMICAL TOXICOLOGY}, author={Melnyk, Lisa Jo and Wang, Zhaohui and Li, Zhilin and Xue, Jianping}, year={2016}, month={Oct}, pages={167–173} }
 @article{zhu_zhang_li_2016, title={The immersed finite volume element method for some interface problems with nonhomogeneous jump conditions}, volume={13}, number={3}, journal={International Journal of Numerical Analysis and Modeling}, author={Zhu, L. and Zhang, Z. Y. and Li, Z. L.}, year={2016}, pages={368–382} }
 @article{zeng_chen_li_2015, title={A parallel Robin-Robin domain decomposition method for H(div)-elliptic problems}, volume={92}, ISSN={["1029-0265"]}, DOI={10.1080/00207160.2014.892587}, abstractNote={In this paper, a parallel Robin–Robin domain decomposition method for H(div)-elliptic problems is proposed. The convergence of the method is proved for both the continuous problem and the finite element approximation. Some numerical testes are also presented to demonstrate the effectiveness of the method.}, number={2}, journal={INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS}, author={Zeng, Yuping and Chen, Jinru and Li, Zhilin}, year={2015}, month={Feb}, pages={394–410} }
 @article{li_xiao_cai_zhao_luo_2015, title={A semi-implicit augmented IIM for Navier-Stokes equations with open, traction, or free boundary conditions}, volume={297}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2015.05.003}, abstractNote={In this paper, a new Navier–Stokes solver based on a finite difference approximation is proposed to solve incompressible flows on irregular domains with open, traction, and free boundary conditions, which can be applied to simulations of fluid structure interaction, implicit solvent model for biomolecular applications and other free boundary or interface problems. For some problems of this type, the projection method and the augmented immersed interface method (IIM) do not work well or does not work at all. The proposed new Navier–Stokes solver is based on the local pressure boundary method, and a semi-implicit augmented IIM. A fast Poisson solver can be used in our algorithm which gives us the potential for developing fast overall solvers in the future. The time discretization is based on a second order multi-step method. Numerical tests with exact solutions are presented to validate the accuracy of the method. Application to fluid structure interaction between an incompressible fluid and a compressible gas bubble is also presented.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Li, Zhilin and Xiao, Li and Cai, Qin and Zhao, Hongkai and Luo, Ray}, year={2015}, month={Sep}, pages={182–193} }
 @article{zhu_zhang_li_2015, title={An immersed finite volume element method for 2D PDEs with discontinuous coefficients and non-homogeneous jump conditions}, volume={70}, ISSN={["1873-7668"]}, DOI={10.1016/j.camwa.2015.04.012}, abstractNote={An immersed finite volume element method is developed to solve 2D elliptic interface problems with a variable coefficient that has a finite jump across an interface. The solution and the flux may also have a finite jump across the interface. Using the source removal technique, an equivalent elliptic interface problem with homogeneous jump conditions is obtained. The nodal basis functions are constructed to satisfy the homogeneous jump conditions near the interface and the usual finite element nodal basis functions are applied away from the interface. The resulting linear problem is simple and easy to solve. A proof of the error estimate in the energy norm is given. Numerical experiments demonstrate the convergence rates of the proposed method with the usual O(h2) in the L2, the L∞ norms, and O(h) in the H1 norm.}, number={2}, journal={COMPUTERS & MATHEMATICS WITH APPLICATIONS}, author={Zhu, Ling and Zhang, Zhiyue and Li, Zhilin}, year={2015}, month={Jul}, pages={89–103} }
 @article{xia_li_ye_2015, title={Effective matrix-free preconditioning for the augmented immersed interface method}, volume={303}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2015.09.050}, abstractNote={We present effective and efficient matrix-free preconditioning techniques for the augmented immersed interface method (AIIM). AIIM has been developed recently and is shown to be very effective for interface problems and problems on irregular domains. GMRES is often used to solve for the augmented variable(s) associated with a Schur complement A in AIIM that is defined along the interface or the irregular boundary. The efficiency of AIIM relies on how quickly the system for A can be solved. For some applications, there are substantial difficulties involved, such as the slow convergence of GMRES (particularly for free boundary and moving interface problems), and the inconvenience in finding a preconditioner (due to the situation that only the products of A and vectors are available). Here, we propose matrix-free structured preconditioning techniques for AIIM via adaptive randomized sampling, using only the products of A and vectors to construct a hierarchically semiseparable matrix approximation to A. Several improvements over existing schemes are shown so as to enhance the efficiency and also avoid potential instability. The significance of the preconditioners includes: (1) they do not require the entries of A or the multiplication of AT with vectors; (2) constructing the preconditioners needs only O(log⁡N) matrix–vector products and O(N) storage, where N is the size of A; (3) applying the preconditioners needs only O(N) flops; (4) they are very flexible and do not require any a priori knowledge of the structure of A. The preconditioners are observed to significantly accelerate the convergence of GMRES, with heuristical justifications of the effectiveness. Comprehensive tests on several important applications are provided, such as Navier–Stokes equations on irregular domains with traction boundary conditions, interface problems in incompressible flows, mixed boundary problems, and free boundary problems. The preconditioning techniques are also useful for several other problems and methods.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Xia, Jianlin and Li, Zhilin and Ye, Xin}, year={2015}, month={Dec}, pages={295–312} }
 @article{zhang_ito_li_zhang_2015, title={Immersed finite elements for optimal control problems of elliptic PDEs with interfaces}, volume={298}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2015.05.050}, abstractNote={This paper presents a numerical method and analysis, based on the variational discretization concept, for optimal control problems governed by elliptic PDEs with interfaces. The method uses a simple uniform mesh which is independent of the interface. Due to the jump of the coefficient across the interface, the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used. Therefore the immersed finite element method (IFEM) developed in Li et al. [20] is used to discretize the state equation required in the variational discretization approach. Optimal error estimates for the control, state and adjoint state are derived. Numerical examples are provided to confirm the theoretical results.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Zhang, Qian and Ito, Kazufumi and Li, Zhilin and Zhang, Zhiyue}, year={2015}, month={Oct}, pages={305–319} }
 @article{li_wang_aspinwall_cooper_kuberry_sanders_zeng_2015, title={Some new analysis results for a class of interface problems}, volume={38}, ISSN={["1099-1476"]}, DOI={10.1002/mma.2865}, abstractNote={Interface problems modeled by differential equations have many applications in mathematical biology, fluid mechanics, material sciences, and many other areas. Typically, interface problems are characterized by discontinuities in the coefficients and/or the Dirac delta function singularities in the source term. Because of these irregularities, solutions to the differential equations are not smooth or discontinuous. In this paper, some new results on the jump conditions of the solution across the interface are derived using the distribution theory and the theory of weak solutions. Some theoretical results on the boundary singularity in which the singular delta function is at the boundary are obtained. Finally, the proof of the convergency of the immersed boundary (IB) method is presented. The IB method is shown to be first‐order convergent in L ∞ norm. Copyright © 2013 John Wiley & Sons, Ltd.}, number={18}, journal={MATHEMATICAL METHODS IN THE APPLIED SCIENCES}, author={Li, Zhilin and Wang, Li and Aspinwall, Eric and Cooper, Racheal and Kuberry, Paul and Sanders, Ashley and Zeng, Ke}, year={2015}, month={Dec}, pages={4530–4539} }
 @article{ruiz alvarez_li_2015, title={The immersed interface method for axis-symmetric problems and application to the Hele-Shaw flow}, volume={264}, ISSN={["1873-5649"]}, DOI={10.1016/j.amc.2015.03.131}, abstractNote={Many physical application problems are axis-symmetric. Using axis-symmetric properties, many three dimensional problems can be solved efficiently using two dimensional axis-symmetric coordinates. In this paper, the immersed interface method (IIM) in axis-symmetric coordinates is developed for elliptic interface problems that have a discontinuous coefficient, solution or flux. A staggered grid is used to overcome the pole singularity. Other challenges include deriving the jump relations in axis-symmetric coordinates, designing the numerical algorithm when the interface is close to the pole (r = 0); computing interface quantities such as the normal and tangential directions, surface derivatives, curvature information, etc. The numerical algorithm is based on a finite difference discretization and uniform grid in the axis-symmetric coordinates. The finite difference scheme is the standard one away from the interface but is modified at grid points near and on the interface. The method is shown to be second order accurate in the infinity norm. The developed new IIM is applied to the Hele-Shaw flow and compared with the results from the literature.}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Ruiz Alvarez, Juan and Li, Zhilin}, year={2015}, month={Aug}, pages={179–197} }
 @article{xu_huang_lai_li_2014, title={A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant}, volume={15}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.241012.310513a}, abstractNote={Abstract In this paper, a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant. The numerical scheme consists of a 3D immersed interface method (IIM) for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface. The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface. This assumption is convenient in conjunction with the level-set techniques. It allows standard Lagrangian interpolation for quantities at the projection points on the interface. The interface jump relations are re-derived accordingly. A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation. Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy. A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.}, number={2}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Xu, Jian-Jun and Huang, Yunqing and Lai, Ming-Chih and Li, Zhilin}, year={2014}, month={Feb}, pages={451–469} }
 @article{ji_chen_li_2014, title={A Symmetric and Consistent Immersed Finite Element Method for Interface Problems}, volume={61}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-014-9837-x}, number={3}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Ji, Haifeng and Chen, Jinru and Li, Zhilin}, year={2014}, month={Dec}, pages={533–557} }
 @article{xiao_cai_li_zhao_luo_2014, title={A multi-scale method for dynamics simulation in continuum solvent models. I: Finite-difference algorithm for Navier–Stokes equation}, volume={616-617}, ISSN={0009-2614}, url={http://dx.doi.org/10.1016/J.CPLETT.2014.10.033}, DOI={10.1016/J.CPLETT.2014.10.033}, abstractNote={A multi-scale framework is proposed for more realistic molecular dynamics simulations in continuum solvent models by coupling a molecular mechanics treatment of solute with a fluid mechanics treatment of solvent. This article reports our initial efforts to formulate the physical concepts necessary for coupling the two mechanics and develop a 3D numerical algorithm to simulate the solvent fluid via the Navier–Stokes equation. The numerical algorithm was validated with multiple test cases. The validation shows that the algorithm is effective and stable, with observed accuracy consistent with our design.}, journal={Chemical Physics Letters}, publisher={Elsevier BV}, author={Xiao, Li and Cai, Qin and Li, Zhilin and Zhao, Hongkai and Luo, Ray}, year={2014}, month={Nov}, pages={67–74} }
 @article{li_2014, title={On convergence of the immersed boundary method for elliptic interface problems}, volume={84}, ISSN={0025-5718 1088-6842}, url={http://dx.doi.org/10.1090/s0025-5718-2014-02932-3}, DOI={10.1090/s0025-5718-2014-02932-3}, abstractNote={Peskin’s Immersed Boundary (IB) method has been one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskin’s IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the $L^{\infty }$ norm. But almost no rigorous proof could be found in the literature until recently [Mori, Comm. Pure. Appl. Math: 61:2008] in which the author showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show first order convergence with a $\log h$ factor of the IB method for elliptic interface problems with Dirichlet boundary conditions.}, number={293}, journal={Mathematics of Computation}, publisher={American Mathematical Society (AMS)}, author={Li, Zhilin}, year={2014}, month={Dec}, pages={1169–1188} }
 @article{li_2014, title={Special issue on fluid structure interactions preface}, volume={7}, number={4}, journal={Numerical Mathematics: Theory, Methods and Applications}, author={Li, Z. L.}, year={2014}, pages={I-} }
 @article{wang_zhang_li_2013, title={A Fourier finite volume element method for solving two-dimensional quasi-geostrophic equations on a sphere}, volume={71}, ISSN={["0168-9274"]}, DOI={10.1016/j.apnum.2013.03.007}, abstractNote={Abstract A new Fourier finite volume element method for solving quasi-geostrophic (QG) equations on a sphere has been developed in this paper. Using the spherical coordinates, a Fourier discretization is used in the longitudinal direction while a finite volume element approximation is used in the latitudinal direction. In our proposed numerical method, the trial and test function spaces are carefully chosen to get accurate approximations. The pole singularity associated with the spherical coordinates is eliminated by changing the resolution near the pole. Some numerical experiments are presented to illustrate accuracy and efficiency of our method and some geostrophic implications of the QG model.}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Wang, Quanxiang and Zhang, Zhiyue and Li, Zhilin}, year={2013}, month={Sep}, pages={1–13} }
 @article{li_song_2013, title={Adaptive mesh refinement techniques for the immersed interface method applied to flow problems}, volume={122}, ISSN={["0045-7949"]}, DOI={10.1016/j.compstruc.2013.03.013}, abstractNote={In this paper, we develop an adaptive mesh refinement strategy of the Immersed Interface Method for flow problems with a moving interface. The work is built on the AMR method developed for two-dimensional elliptic interface problems in the paper [12] (CiCP, 12(2012), 515–527). The interface is captured by the zero level set of a Lipschitz continuous function φ(x, y, t). Our adaptive mesh refinement is built within a small band of ∣φ(x, y, t)∣ ⩽ δ with finer Cartesian meshes. The AMR-IIM is validated for Stokes and Navier–Stokes equations with exact solutions, moving interfaces driven by the surface tension, and classical bubble deformation problems. A new simple area preserving strategy is also proposed in this paper for the level set method.}, journal={COMPUTERS & STRUCTURES}, author={Li, Zhilin and Song, Peng}, year={2013}, month={Jun}, pages={249–258} }
 @article{liu_wang_wang_li_zhao_luo_2013, title={Exploring a charge-central strategy in the solution of Poisson's equation for biomolecular applications}, volume={15}, ISSN={["1463-9084"]}, DOI={10.1039/c2cp41894k}, abstractNote={Continuum solvent treatments based on the Poisson–Boltzmann equation have been widely accepted for energetic analysis of biomolecular systems. In these approaches, the molecular solute is treated as a low dielectric region and the solvent is treated as a high dielectric continuum. The existence of a sharp dielectric jump at the solute–solvent interface poses a challenge to model the solvation energetics accurately with such a simple mathematical model. In this study, we explored and evaluated a strategy based on the “induced surface charge” to eliminate the dielectric jump within the finite-difference discretization scheme. In addition to the use of the induced surface charges in solving the equation, the second-order accurate immersed interface method is also incorporated to discretize the equation. The resultant linear system is solved with the GMRES algorithm to explicitly impose the flux conservation condition across the solvent–solute interface. The new strategy was evaluated on both analytical and realistic biomolecular systems. The numerical tests demonstrate the feasibility of utilizing induced surface charge in the finite-difference solution of the Poisson–Boltzmann equation. The analysis data further show that the strategy is consistent with theory and the classical finite-difference method on the tested systems. Limitations of the current implementations and further improvements are also analyzed and discussed to fully bring out its potential of achieving higher numerical accuracy.}, number={1}, journal={PHYSICAL CHEMISTRY CHEMICAL PHYSICS}, author={Liu, Xingping and Wang, Changhao and Wang, Jun and Li, Zhilin and Zhao, Hongkai and Luo, Ray}, year={2013}, pages={129–141} }
 @article{wang_wang_cai_li_zhao_luo_2013, title={Exploring accurate Poisson-Boltzmann methods for biomolecular simulations}, volume={1024}, ISSN={["1872-7999"]}, DOI={10.1016/j.comptc.2013.09.021}, abstractNote={Accurate and efficient treatment of electrostatics is a crucial step in computational analyses of biomolecular structures and dynamics. In this study, we have explored a second-order finite-difference numerical method to solve the widely used Poisson–Boltzmann equation for electrostatic analyses of realistic biomolecules. The so-called immersed interface method was first validated and found to be consistent with the classical weighted harmonic averaging method for a diversified set of test biomolecules. The numerical accuracy and convergence behaviors of the new method were next analyzed in its computation of numerical reaction field grid potentials, energies, and atomic solvation forces. Overall similar convergence behaviors were observed as those by the classical method. Interestingly, the new method was found to deliver more accurate and better-converged grid potentials than the classical method on or nearby the molecular surface, though the numerical advantage of the new method is reduced when grid potentials are extrapolated to the molecular surface. Our exploratory study indicates the need for further improving interpolation/extrapolation schemes in addition to the developments of higher-order numerical methods that have attracted most attention in the field.}, journal={COMPUTATIONAL AND THEORETICAL CHEMISTRY}, author={Wang, Changhao and Wang, Jun and Cai, Qin and Li, Zhilin and Zhao, Hong-Kai and Luo, Ray}, year={2013}, month={Nov}, pages={34–44} }
 @article{botello-smith_liu_cai_li_zhao_luo_2013, title={Numerical Poisson-Boltzmann model for continuum membrane systems}, volume={555}, ISSN={["1873-4448"]}, DOI={10.1016/j.cplett.2012.10.081}, abstractNote={Membrane protein systems are important computational research topics due to their roles in rational drug design. In this study, we developed a continuum membrane model utilizing a level set formulation under the numerical Poisson-Boltzmann framework within the AMBER molecular mechanics suite for applications such as protein-ligand binding affinity and docking pose predictions. Two numerical solvers were adapted for periodic systems to alleviate possible edge effects. Validation on systems ranging from organic molecules to membrane proteins up to 200 residues, demonstrated good numerical properties. This lays foundations for sophisticated models with variable dielectric treatments and second-order accurate modeling of solvation interactions.}, journal={CHEMICAL PHYSICS LETTERS}, author={Botello-Smith, Wesley M. and Liu, Xingping and Cai, Qin and Li, Zhilin and Zhao, Hongkai and Luo, Ray}, year={2013}, month={Jan}, pages={274–281} }
 @article{song_xue_li_2013, title={Simulation of Longitudinal Exposure Data with Variance-Covariance Structures Based on Mixed Models}, volume={33}, ISSN={["0272-4332"]}, DOI={10.1111/j.1539-6924.2012.01869.x}, abstractNote={Longitudinal data are important in exposure and risk assessments, especially for pollutants with long half-lives in the human body and where chronic exposures to current levels in the environment raise concerns for human health effects. It is usually difficult and expensive to obtain large longitudinal data sets for human exposure studies. This article reports a new simulation method to generate longitudinal data with flexible numbers of subjects and days. Mixed models are used to describe the variance-covariance structures of input longitudinal data. Based on estimated model parameters, simulation data are generated with similar statistical characteristics compared to the input data. Three criteria are used to determine similarity: the overall mean and standard deviation, the variance components percentages, and the average autocorrelation coefficients. Upon the discussion of mixed models, a simulation procedure is produced and numerical results are shown through one human exposure study. Simulations of three sets of exposure data successfully meet above criteria. In particular, simulations can always retain correct weights of inter- and intrasubject variances as in the input data. Autocorrelations are also well followed. Compared with other simulation algorithms, this new method stores more information about the input overall distribution so as to satisfy the above multiple criteria for statistical targets. In addition, it generates values from numerous data sources and simulates continuous observed variables better than current data methods. This new method also provides flexible options in both modeling and simulation procedures according to various user requirements.}, number={3}, journal={RISK ANALYSIS}, author={Song, Peng and Xue, Jianping and Li, Zhilin}, year={2013}, month={Mar}, pages={469–479} }
 @article{hou_li_wang_wang_2012, title={A Numerical Method for Solving Elasticity Equations with Interfaces}, volume={12}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.160910.130711s}, abstractNote={Abstract Solving elasticity equations with interfaces is a challenging problem for most existing methods. Nonetheless, it has wide applications in engineering and science. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L ∞ norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner.}, number={2}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Hou, Songming and Li, Zhilin and Wang, Liqun and Wang, Wei}, year={2012}, month={Aug}, pages={595–612} }
 @article{ho_li_lubkin_2012, title={AN AUGMENTED IMMERSED INTERFACE METHOD FOR MOVING STRUCTURES WITH MASS}, volume={17}, ISSN={["1531-3492"]}, url={http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=ORCID&SrcApp=OrcidOrg&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=WOS:000301177800006&KeyUID=WOS:000301177800006}, DOI={10.3934/dcdsb.2012.17.1175}, abstractNote={We present an augmented immersed interface method for simulating the dynamics of a deformable structure with mass in an incompressible fluid. The fluid is modeled by the Navier-Stokes equations in two dimensions. The acceleration of the structure due to mass is coupled with the flow velocity and the pressure. The surface tension of the structure is assumed to be a constant for simplicity. In our method, we treat the unknown acceleration as the only augmented variable so that the augmented immersed interface method can be applied. We use a modified projection method that can enforce the pressure jump conditions corresponding to the unknown acceleration. The acceleration must match the flow acceleration along the interface. The proposed augmented method is tested against an exact solution with a stationary interface. It shows that the augmented method has a second order of convergence in space. The dynamics of a deformable circular structure with mass is also investigated. It shows that the fluid-structure system has bi-stability: a stationary state for a smaller Reynolds number and an oscillatory state for a larger Reynolds number. The observation agrees with those in the literature.}, number={4}, journal={DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B}, author={Ho, Jian and Li, Zhilin and Lubkin, Sharon R.}, year={2012}, month={Jun}, pages={1175–1184} }
 @article{li_song_2012, title={An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods}, volume={12}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.070211.150811s}, abstractNote={An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources. The interface is represented by the zero level set of a Lipschitz function φ(x,y). Our adaptive mesh refinement is done within a small tube of |φ(x,y)|≤ δ with finer Cartesian meshes. The discrete linear system of equations is solved by a multigrid solver. The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically, therefore, reduce the size of the linear system of the equations. Numerical examples presented show the efficiency of the grid refinement strategy.}, number={2}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Li, Zhilin and Song, Peng}, year={2012}, month={Aug}, pages={515–527} }
 @book{li_2012, title={Discrete and Continuous Dynamical Systems, Series B: Mathematical Modeling, Analysis and Computations}, volume={17}, number={4}, year={2012}, month={Jun} }
 @article{caraus_li_2012, title={Numerical Solutions of the System of Singular Integro-Differential Equations in Classical Holder Spaces}, volume={4}, ISSN={["2070-0733"]}, DOI={10.4208/aamm.12-12s04}, abstractNote={Abstract New numerical methods based on collocation methods with the mechanical quadrature rules are proposed to solve some systems of singular integro-differential equations that are defined on arbitrary smooth closed contours of the complex plane. We carry out the convergence analysis in classical Hölder spaces. A numerical example is also presented.}, number={6}, journal={ADVANCES IN APPLIED MATHEMATICS AND MECHANICS}, author={Caraus, Iurie and Li, Zhilin}, year={2012}, month={Dec}, pages={737–750} }
 @article{wan_li_2012, title={SOME NEW FINITE DIFFERENCE METHODS FOR HELMHOLTZ EQUATIONS ON IRREGULAR DOMAINS OR WITH INTERFACES}, volume={17}, ISSN={["1531-3492"]}, DOI={10.3934/dcdsb.2012.17.1155}, abstractNote={Solving a Helmholtz equation $\Delta u + \lambda u = f$efficiently is a challenge for manyapplications. For example, the core part of many efficient solvers for theincompressible Navier-Stokes equations is to solve one or severalHelmholtz equations. In this paper, two new finite difference methodsare proposed for solving Helmholtz equations on irregular domains, orwith interfaces. For Helmholtz equations on irregular domains, theaccuracy of the numerical solution obtained using the existing augmentedimmersed interface method (AIIM) may deteriorate when the magnitude of$\lambda$is large. In our new method, we use a level set function to extendthe source term and the PDE to a larger domain before we apply the AIIM.For Helmholtz equations with interfaces,a new maximum principle preserving finite difference method is developed.The new method still uses the standard five-point stencil with modificationsof the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite differenceequations satisfies the signproperty of the discrete maximum principle and can be solved efficientlyusing a multigrid solver. The finite difference method is also extended tohandle temporal discretized equations where the solution coefficient $\lambda$ isinversely proportional to the mesh size.}, number={4}, journal={DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B}, author={Wan, Xiaohai and Li, Zhilin}, year={2012}, month={Jun}, pages={1155–1174} }
 @book{special issue on fluid dynamics, analysis and numerics_2012, year={2012} }
 @article{feng_li_2012, title={Simplified immersed interface methods for elliptic interface problems with straight interfaces}, volume={28}, ISSN={0749-159X}, url={http://dx.doi.org/10.1002/num.20614}, DOI={10.1002/num.20614}, abstractNote={In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one-dimensional problems or two-dimensional problems with circular interfaces, we propose a conservative second-order finite difference scheme whose coefficient matrix is symmetric and definite. For two-dimensional problems with straight interfaces, we first propose a conservative first-order finite difference scheme, then use the Richardson extrapolation technique to get a second-order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 188–203, 2012}, number={1}, journal={Numerical Methods for Partial Differential Equations}, publisher={Wiley}, author={Feng, Xiufang and Li, Zhilin}, year={2012}, month={Jan}, pages={188–203} }
 @book{special issue on fluid motion driven by immersed structures_2012, url={https://www.researchgate.net/publication/296762774_Special_Issue_on_Fluid_Motion_Driven_by_Immersed_Structures_Preface}, journal={Global Sciences}, year={2012} }
 @article{layton_stockie_li_huang_2012, title={Special issue on fluid motion driven by immersed structures preface}, volume={12}, number={2}, journal={Communications in Computational Physics}, author={Layton, A. and Stockie, J. and Li, Z. L. and Huang, H. X.}, year={2012}, pages={I-} }
 @article{ito_li_qiao_2012, title={The Sensitivity Analysis for the Flow Past Obstacles Problem with Respect to the Reynolds Number}, volume={4}, ISSN={["2070-0733"]}, DOI={10.4208/aamm.11-m1110}, abstractNote={Abstract In this paper, numerical sensitivity analysis with respect to the Reynolds number for the flow past obstacle problem is presented. To carry out such analysis, at each time step, we need to solve the incompressible Navier-Stokes equations on irregular domains twice, one for the primary variables; the other is for the sensitivity variables with homogeneous boundary conditions. The Navier-Stokes solver is the augmented immersed interface method for Navier-Stokes equations on irregular domains. One of the most important contribution of this paper is that our analysis can predict the critical Reynolds number at which the vortex shading begins to develop in the wake of the obstacle. Some interesting experiments are shown to illustrate how the critical Reynolds number varies with different geometric settings.}, number={1}, journal={ADVANCES IN APPLIED MATHEMATICS AND MECHANICS}, author={Ito, Kazufumi and Li, Zhilin and Qiao, Zhonghua}, year={2012}, month={Feb}, pages={21–35} }
 @article{xie_li_qiao_2011, title={A finite element method for elasticity interface problems with locally modified triangulations}, volume={8}, number={2}, journal={International Journal of Numerical Analysis and Modeling}, author={Xie, H. and Li, Z. L. and Qiao, Z. H.}, year={2011}, pages={189–200} }
 @article{wu_li_lai_2011, title={Adaptive mesh refinement for elliptic interface problems using the non-conforming immersed finite element method}, volume={8}, number={3}, journal={International Journal of Numerical Analysis and Modeling}, author={Wu, C. T. and Li, Z. L. and Lai, M. C.}, year={2011}, pages={466–483} }
 @article{feng_li_qiao_2011, title={High order compact finite difference schemes for the helmholtz equation with discontinuous coefficients}, volume={29}, number={3}, journal={Journal of Computational Mathematics}, author={Feng, X. F. and Li, Z. L. and Qiao, Z. H.}, year={2011}, pages={324–340} }
 @article{xu_li_lowengrub_zhao_2011, title={Numerical Study of Surfactant-Laden Drop-Drop Interactions}, volume={10}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.090310.020610a}, abstractNote={Abstract In this paper, we numerically investigate the effects of surfactant on drop-drop interactions in a 2D shear flow using a coupled level-set and immersed interface approach proposed in (Xu et al., J. Comput. Phys., 212 (2006), 590-616). We find that surfactant plays a critical and nontrivial role in drop-drop interactions. In particular, we find that the minimum distance between the drops is a non-monotone function of the surfactant coverage and Capillary number. This non-monotonic behavior, which does not occur for clean drops, is found to be due to the presence of Marangoni forces along the drop interfaces. This suggests that there are non-monotonic conditions for coalescence of surfactant-laden drops, as observed in recent experiments of Leal and co-workers. Although our study is two-dimensional, we believe that drop-drop interactions in three-dimensional flows should be qualitatively similar as the Maragoni forces in the near contact region in 3D should have a similar effect.}, number={2}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Xu, Jian-Jun and Li, Zhilin and Lowengrub, John and Zhao, Hongkai}, year={2011}, month={Aug}, pages={453–473} }
 @article{ruiz alvarez_chen_li_2011, title={The IIM in polar coordinates and its application to electro capacitance tomography problems}, volume={57}, ISSN={["1572-9265"]}, DOI={10.1007/s11075-010-9436-3}, number={3}, journal={NUMERICAL ALGORITHMS}, author={Ruiz Alvarez, Juan and Chen, Jinru and Li, Zhilin}, year={2011}, month={Jul}, pages={405–423} }
 @article{li_lai_he_zhao_2010, title={An augmented method for free boundary problems with moving contact lines}, volume={39}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2010.01.013}, abstractNote={An augmented immersed interface method (IIM) is proposed for simulating one-phase moving contact line problems in which a liquid drop spreads or recoils on a solid substrate. While the present two-dimensional mathematical model is a free boundary problem, in our new numerical method, the fluid domain enclosed by the free boundary is embedded into a rectangular one so that the problem can be solved by a regular Cartesian grid method. We introduce an augmented variable along the free boundary so that the stress balancing boundary condition is satisfied. A hybrid time discretization is used in the projection method for better stability. The resultant Helmholtz/Poisson equations with interfaces then are solved by the IIM in an efficient way. Several numerical tests including an accuracy check, and the spreading and recoiling processes of a liquid drop are presented in detail.}, number={6}, journal={COMPUTERS & FLUIDS}, author={Li, Zhilin and Lai, Ming-Chih and He, Guowei and Zhao, Hongkai}, year={2010}, month={Jun}, pages={1033–1040} }
 @article{gong_li_2010, title={Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions}, volume={3}, number={1}, journal={Numerical Mathematics: Theory, Methods and Applications}, author={Gong, Y. and Li, Z. L.}, year={2010}, pages={23–39} }
 @article{yang_zhang_li_he_2009, title={A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations}, volume={228}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2009.07.023}, abstractNote={The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.}, number={20}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Yang, Xiaolei and Zhang, Xing and Li, Zhilin and He, Guo-Wei}, year={2009}, month={Nov}, pages={7821–7836} }
 @article{ito_lai_li_2009, title={A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains}, volume={228}, ISSN={["0021-9991"]}, DOI={10.1016/j.jcp.2008.12.028}, abstractNote={An augmented method based on a Cartesian grid is proposed for the incompressible Navier–Stokes equations in irregular domains. The irregular domain is embedded into a rectangular one so that a fast Poisson solver can be utilized in the projection method. Unlike several methods suggested in the literature that set the force strengths as unknowns, which often results in an ill-conditioned linear system, we set the jump in the normal derivative of the velocity as the augmented variable. The new approach improves the condition number of the system for the augmented variable significantly. Using the immersed interface method, we are able to achieve second order accuracy for the velocity. Numerical results and comparisons to benchmark tests are given to validate the new method. A lid-driven cavity flow with multiple obstacles and different geometries are also presented.}, number={7}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Ito, Kazufumi and Lai, Ming-Chih and Li, Zhilin}, year={2009}, month={Apr}, pages={2616–2628} }
 @article{wang_cai_li_zhao_luo_2009, title={Achieving energy conservation in Poisson-Boltzmann molecular dynamics: Accuracy and precision with finite-difference algorithms}, volume={468}, ISSN={["1873-4448"]}, DOI={10.1016/j.cplett.2008.12.049}, abstractNote={Violation of energy conservation in Poisson–Boltzmann molecular dynamics, due to the limited accuracy and precision of numerical methods, is a major bottleneck preventing its wide adoption in biomolecular simulations. We explored the ideas of enforcing interface conditions by the immerse interface method and of removing charge singularity to improve the finite-difference methods. Our analysis of these ideas on an analytical test system shows significant improvement in both energies and forces. Our analysis further indicates the need for more accurate force calculation, especially the boundary force calculation.}, number={4-6}, journal={CHEMICAL PHYSICS LETTERS}, author={Wang, Jun and Cai, Qin and Li, Zhi-Lin and Zhao, Hong-Kai and Luo, Ray}, year={2009}, month={Jan}, pages={112–118} }
 @article{wang_chen_xu_li_2009, title={An additive Schwarz preconditioner for the mortar-type rotated Q(1) FEM for elliptic problems with discontinuous coefficients}, volume={59}, ISSN={["1873-5460"]}, DOI={10.1016/j.apnum.2008.11.006}, abstractNote={In this paper, we propose an additive Schwarz preconditioner for the mortar-type rotated Q1 finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. The work here is an extension of the research presented in [L. Marcinkowski, Additive Schwarz method for mortar discretization of elliptic problems with P1 non-conforming elements, BIT 45 (2005) 375–394]. Our analysis is valid for rectangular or L-shaped domains, which are partitioned by rectangular subdomains and meshes. We have shown that our proposed method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is O((1+log(H/h))2), which is independent of the jump in the coefficient. Numerical experiments presented in this paper have confirmed our theoretical analysis.}, number={7}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Wang, Feng and Chen, Jinru and Xu, Wei and Li, Zhilin}, year={2009}, month={Jul}, pages={1657–1667} }
 @inproceedings{jiang_li_lubkin_2009, title={Analysis and computation for a fluid mixture model}, volume={5}, number={2-4}, booktitle={Communications in Computational Physics}, author={Jiang, Q. L. and Li, Z. L. and Lubkin, S. R.}, year={2009}, pages={620–634} }
 @book{khoo_li_lin_2009, place={Singapore; Hackensack, NJ}, series={Lecture Notes Series (National University of Singapore, Institute for Mathematical Sciences)}, title={Interface Problems and Methods in Biological and Physical Flows}, ISBN={9789812837844 9789812837851}, ISSN={1793-0758}, url={http://dx.doi.org/10.1142/7147}, DOI={10.1142/7147}, abstractNote={An Introduction to the Immersed Boundary and the Immersed Interface Methods Lecture Notes on Nonlinear Tumor Growth: Modeling and Simulation Progress in Modeling Pulsed Detonations Direct Numerical Simulations of Multiphase Flows.}, journal={WORLD SCIENTIFIC}, publisher={World Scientific}, author={Khoo, Boo Cheong and Li, Zhilin and Lin, Ping}, editor={Khoo, Boo Cheong and Li, Zhilin and Lin, PingEditors}, year={2009}, month={May}, collection={Lecture Notes Series (National University of Singapore, Institute for Mathematical Sciences)} }
 @book{b. c. khoo_lin_2009, title={Interface problems and methods in biological and physical flows}, ISBN={9789812837844}, publisher={New Jersey: World Scientific}, year={2009} }
 @article{chen_li_lin_2008, title={A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow}, volume={29}, ISSN={["1572-9044"]}, DOI={10.1007/s10444-007-9043-6}, number={2}, journal={ADVANCES IN COMPUTATIONAL MATHEMATICS}, author={Chen, Guo and Li, Zhilin and Lin, Ping}, year={2008}, month={Aug}, pages={113–133} }
 @inproceedings{xie_ito_li_toivanen_2008, title={A finite element method for interface problems with locally modified triangulations}, volume={466}, DOI={10.1090/conm/466/09122}, booktitle={Moving interface problems and applications in fluid dynamics}, author={Xie, H. and Ito, K. and Li, Zhilin and Toivanen, J.}, year={2008}, pages={179–190} }
 @article{rutka_li_2008, title={An explicit jump immersed interface method for two-phase Navier-Stokes equations with interfaces}, volume={197}, ISSN={["1879-2138"]}, DOI={10.1016/j.cma.2007.12.016}, abstractNote={In this paper, we propose an explicit jump immersed interface method (EJIIM) for the incompressible Navier–Stokes equations with a discontinuous viscosity and singular forces along one or several interfaces in the solution domain. The EJIIM is used to get a second-order finite difference discretization at the grid points near or on the interface even if the jump conditions for the two-phase flow are complicated. The new method is based on a projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully second-order accuracy for the velocity and nearly second-order accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. The work here is a necessary first step in developing second-order accurate algorithm for two-phase Navier–Stokes equations with a moving interface.}, number={25-28}, journal={COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING}, author={Rutka, Vita and Li, Zhilin}, year={2008}, pages={2317–2328} }
 @article{tan_le_li_lim_khoo_2008, title={An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane}, volume={227}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2008.08.013}, abstractNote={This paper presents an implementation of the second-order accurate immersed interface method to simulate the motion of the flexible elastic membrane immersed in two viscous incompressible fluids with different viscosities, which further develops the work reported in Tan et al. [Z.-J. Tan, D.V. Le, K.M. Lim, B.C. Khoo, An Immersed Interface Method for the Incompressible Navier–Stokes Equations with Discontinuous Viscosity Across the Interface, submitted for publication] focussing mainly on the fixed interface problems. In this work, we introduce the velocity components at the membrane as two augmented unknown interface variables to decouple the originally coupled jump conditions for the velocity and pressure. Three forms of augmented equation are derived to determine the augmented variables to satisfy the continuous condition of the velocity. The velocity at the membrane, which determine the motion of the membrane, is then solved by the GMRES iterative method. The forces calculated from the configuration of the flexible elastic membrane and the augmented variables are interpolated using cubic splines and applied to the fluid through the jump conditions. The position of the flexible elastic membrane is updated implicitly using a quasi-Newton method (BFGS) within each time step. The Navier–Stokes equations are solved on a staggered Cartesian grid using a second order accurate projection method with the incorporation of spatial and temporal jump conditions. In addition, we also show that the inclusion of the temporal jump contributions has non-negligible effect on the simulation results when the grids are crossed by the membrane. Using the above method, we assess the effect of different viscosities on the flow solution and membrane motion.}, number={23}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Tan, Zhijun and Le, D. V. and Li, Zhilin and Lim, K. M. and Khoo, B. C.}, year={2008}, month={Dec}, pages={9955–9983} }
 @article{gong_li_li_2008, title={Immersed-interface finite-element methods for elliptic interface problems with nonhomogeneous jump conditions}, volume={46}, ISSN={["1095-7170"]}, DOI={10.1137/060666482}, abstractNote={In this work, a class of new finite-element methods, called immersed-interface finite-element methods, is developed to solve elliptic interface problems with nonhomogeneous jump conditions. Simple non-body-fitted meshes are used. A single function that satisfies the same nonhomogeneous jump conditions is constructed using a level-set representation of the interface. With such a function, the discontinuities across the interface in the solution and flux are removed, and an equivalent elliptic interface problem with homogeneous jump conditions is formulated. Special finite-element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions. Error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate. These methods are designed as an efficient component of the finite-element level-set methodology for fast simulation of interface dynamics that does not require remeshing.}, number={1}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Gong, Yan and Li, Bo and Li, Zhilin}, year={2008}, pages={472–495} }
 @article{wan_li_lubkin_2008, title={Mechanics of mesenchymal contribution to clefting force in branching morphogenesis}, volume={7}, ISSN={["1617-7959"]}, url={http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=ORCID&SrcApp=OrcidOrg&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=WOS:000258612200008&KeyUID=WOS:000258612200008}, DOI={10.1007/s10237-007-0105-y}, number={5}, journal={BIOMECHANICS AND MODELING IN MECHANOBIOLOGY}, author={Wan, Xiaohai and Li, Zhilin and Lubkin, Sharon R.}, year={2008}, month={Oct}, pages={417–426} }
 @article{li_pao_qiao_2007, title={A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations}, volume={30}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-005-9019-y}, DOI={10.1007/s10915-005-9019-y}, number={1}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Li, Zhilin and Pao, C. V. and Qiao, Zhonghua}, year={2007}, month={Jan}, pages={61–81} }
 @article{gremaud_kuster_li_2007, title={A study of numerical methods for the level set approach}, volume={57}, ISSN={["0168-9274"]}, DOI={10.1016/j.apnum.2006.07.022}, abstractNote={The computation of moving curves by the level set method typically requires reinitializations of the underlying level set function. Two types of reinitialization methods are studied: a high order “PDE” approach and a second order Fast Marching method. Issues related to the efficiency and implementation of both types of methods are discussed, with emphasis on the tube/narrow band implementation and accuracy considerations. The methods are also tested and compared. Fast Marching reinitialization schemes are faster but limited to second order, PDE based reinitialization schemes can easily be made more accurate but are slower, even with a tube/narrow band implementation.}, number={5-7}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Gremaud, Pierre A. and Kuster, Christopher M. and Li, Zhilin}, year={2007}, pages={837–846} }
 @article{li_ito_lai_2007, title={An augmented approach for Stokes equations with a discontinuous viscosity and singular forces}, volume={36}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2006.03.003}, abstractNote={For Stokes equations with a discontinuous viscosity across an arbitrary interface or/and singular forces along the interface, it is known that the pressure is discontinuous and the velocity is non-smooth. It has been shown that these discontinuities are coupled together, which makes it difficult to obtain accurate numerical solutions. In this paper, a new numerical method that decouples the jump conditions of the fluid variables through two augmented variables has been developed. The GMRES iterative method is used to solve the Schur complement system for the augmented variables that are only defined on the interface. The augmented approach also rescales the Stokes equations in such a way that a fast Poisson solver can be used in each iteration. Numerical tests using examples that have analytic solutions show that the new method has average second order accuracy for the velocity in the infinity norm. An example of a moving interface problem is also presented.}, number={3}, journal={COMPUTERS & FLUIDS}, author={Li, Zhilin and Ito, Kazufumi and Lai, Ming-Chih}, year={2007}, month={Mar}, pages={622–635} }
 @article{li_lubkin_wan_2007, title={An augmented immersed interface-level set method for Stokes equations with discontinuous viscosity}, volume={15}, ISSN={1072-6691}, journal={Electronic Journal of Differential Equations, Conference}, author={Li, Zhilin and Lubkin, S. and Wan, X.}, year={2007}, pages={193–210} }
 @article{li_lai_ito_2007, title={An immersed interface method for the Navier-Stokes equations on irregular domains}, volume={7}, ISSN={1617-7061 1617-7061}, url={http://dx.doi.org/10.1002/pamm.200700758}, DOI={10.1002/pamm.200700758}, abstractNote={Abstract An augmented method based on a Cartesian grid is proposed for the incompressible Navier Stokes equations on an irregular domain. The irregular domain is embedded into a rectangular one so that a fast Poisson solver can be used in the projection method. Unlike several methods suggested in the literature that set the force strengths as unknowns, which often results an ill‐conditioned linear system, we set the jump in the normal derivative of the velocity as the augmented variable. The new approach improve the condition number of the system for the augmented variable significantly. Using the immersed interface method, we achieve second order accuracy for the velocity. Numerical results and comparisons are given to validate the new method. Some interesting new numerical experiments results are also presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)}, number={1}, journal={PAMM}, publisher={Wiley}, author={Li, Zhilin and Lai, Ming-Chih and Ito, Kazufumi}, year={2007}, month={Dec}, pages={1025401–1025402} }
 @book{li_2007, title={Applied Numerical Mathematics for the International Conference on Scientific Computing}, volume={57}, number={5-7}, journal={Contemporary Mathematics}, publisher={American Mathematical Society}, year={2007} }
 @book{moving interface problems and applications in fluid dynamics_2007, url={https://books.google.com/books/about/Moving_Interface_Problems_and_Applicatio.html?id=fMYbCAAAQBAJ}, journal={AMS Contemporary mathematics 466}, year={2007} }
 @article{li_song_tang_2007, title={Preface of the special issue of APNUM - International Conference on Scientific Computing in Nanjing, China}, volume={57}, ISSN={["0168-9274"]}, DOI={10.1016/j.apnum.2006.07.023}, number={5-7}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Li, Zhilin and Song, Yongzhong and Tang, Tao}, year={2007}, pages={473–474} }
 @book{special issue for the international conference on scientific computing_2007, url={https://www.sciencedirect.com/journal/applied-numerical-mathematics/vol/57/issue/5}, year={2007}, month={May} }
 @article{xu_li_lowengrub_zhao_2006, title={A level-set method for interfacial flows with surfactant}, volume={212}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2005.07.016}, abstractNote={A level-set method for the simulation of fluid interfaces with insoluble surfactant is presented in two-dimensions. The method can be straightforwardly extended to three-dimensions and to soluble surfactants. The method couples a semi-implicit discretization for solving the surfactant transport equation recently developed by Xu and Zhao [J. Xu, H. Zhao. An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput. 19 (2003) 573–594] with the immersed interface method originally developed by LeVeque and Li and [R. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31 (1994) 1019–1044] for solving the fluid flow equations and the Laplace–Young boundary conditions across the interfaces. Novel techniques are developed to accurately conserve component mass and surfactant mass during the evolution. Convergence of the method is demonstrated numerically. The method is applied to study the effects of surfactant on single drops, drop–drop interactions and interactions among multiple drops in Stokes flow under a steady applied shear. Due to Marangoni forces and to non-uniform Capillary forces, the presence of surfactant results in larger drop deformations and more complex drop–drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the level-set method has been used to simulate fluid interfaces with surfactant.}, number={2}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Xu, JJ and Li, ZL and Lowengrub, J and Zhao, HK}, year={2006}, month={Mar}, pages={590–616} }
 @article{li_wan_ito_lubkin_2006, title={An augmented approach for the pressure boundary condition in a Stokes flow}, volume={1}, number={5}, journal={Communications in Computational Physics}, author={Li, Z. L. and Wan, X. H. and Ito, K. and Lubkin, S. R.}, year={2006}, pages={874–885} }
 @article{li_qiao_tang_2006, title={Efficient numerical methods for the 2D nonlinear Poisson–Boltzmann equation modeling charged spheres}, volume={24}, number={3}, journal={Journal of Computational Mathematics}, author={Li, Zhilin and Qiao, Zhong-hua and Tang, Tao}, year={2006}, pages={252–264} }
 @article{lai_li_lin_2006, title={Fast solvers for 3D Poisson equations involving interfaces in a finite or the infinite domain}, volume={191}, ISSN={["1879-1778"]}, DOI={10.1016/j.cam.2005.04.025}, abstractNote={In this paper, numerical methods are proposed for Poisson equations defined in a finite or infinite domain in three dimensions. In the domain, there can exists an interface across which the source term, the flux, and therefore the solution may be discontinuous. The existence and uniqueness of the solution are also discussed. To deal with the discontinuity in the source term and in the flux, the original problem is transformed to a new one with a smooth solution. Such a transformation can be carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three-dimensional function. An auxiliary sphere is used to separate the infinite region into an interior and exterior domain. The Kelvin's inversion is used to map the exterior domain into an interior domain. The two Poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously. By choosing the mesh size carefully and exploiting the fast Fourier transform, the resulting finite difference equations can be solved efficiently. The approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain. Numerical results demonstrate second order accuracy of our algorithms.}, number={1}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Lai, MC and Li, ZL and Lin, XB}, year={2006}, month={Jun}, pages={106–125} }
 @article{ito_li_2006, title={Interface conditions for Stokes equations with a discontinuous viscosity and surface sources}, volume={19}, ISSN={["0893-9659"]}, DOI={10.1016/j.aml.2005.02.041}, abstractNote={The interface conditions, or jump conditions, for the pressure and the velocity of the solution to the incompressible Stokes equations with a discontinuous viscosity and a singular source along an interface are derived in this work. While parts of the results agree with those in the literature, some of the results are new. These theoretical results are useful for developing accurate numerical methods for the interface problem.}, number={3}, journal={APPLIED MATHEMATICS LETTERS}, author={Ito, K and Li, ZL}, year={2006}, month={Mar}, pages={229–234} }
 @book{li_ito_2006, title={The immersed interface method: Numerical solutions of PDEs involving interfaces and irregular domains}, ISBN={0898716098}, DOI={10.1137/1.9780898717464}, publisher={Philadelphia: SIAM, Society for Industrial and Applied Mathematics}, author={Li, Zhilin and Ito, K.}, year={2006} }
 @inbook{li_2005, place={Berlin}, series={Lecture Notes in Computer Science}, title={Augmented Strategies for Interface and Irregular Domain Problems}, volume={3401}, ISBN={9783540249375 9783540318521}, ISSN={0302-9743 1611-3349}, url={http://dx.doi.org/10.1007/978-3-540-31852-1_7}, DOI={10.1007/978-3-540-31852-1_7}, booktitle={Numerical Analysis and Its Applications}, publisher={Springer}, author={Li, Zhilin}, editor={Li, Z. and Vulkov, L. and Wasniewski, J.Editors}, year={2005}, pages={66–79}, collection={Lecture Notes in Computer Science} }
 @article{ito_li_kyei_2005, title={Higher-order, Cartesian grid based finite difference schemes for elliptic equations on irregular domains}, volume={27}, ISSN={["1095-7197"]}, DOI={10.1137/03060120X}, abstractNote={Second and fourth order Cartesian grid based finite difference methods are proposed for elliptic and parabolic partial differential equations, and associated eigenvalue problems on irregular domains with general boundary conditions. Our methods are based on the continuation of a solution idea using multivariable Taylor's expansion of the solution about selected boundary points, and the core ideas of the immersed interface method. The methods offer systematic treatment of the general boundary conditions in two- and three-dimensional domains and are directly applied to semi-discretize heat equations on irregular domains. Convergence analysis and numerical examples are presented. The validity and effectiveness of the proposed methods are demonstrated through our numerical results including computations of the eigenvalues of the associated eigenvalue problem.}, number={1}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Ito, K and Li, ZL and Kyei, Y}, year={2005}, pages={346–367} }
 @article{li_yang_2005, title={Immersed finite element for elasticity system with discontinuities}, volume={383}, journal={AMS Contemporary Mathematics}, author={Li, Zhilin and Yang, X.}, year={2005}, pages={285–298} }
 @book{li_vulkov_waśniewski_2005, place={Berlin}, series={Lecture Notes in Computer Science}, title={Numerical Analysis and Its Applications}, ISBN={9783540249375 9783540318521}, ISSN={0302-9743 1611-3349}, url={http://dx.doi.org/10.1007/b106395}, DOI={10.1007/b106395}, publisher={Springer}, year={2005}, collection={Lecture Notes in Computer Science} }
 @article{li_lin_lin_rogers_2004, title={An immersed finite element space and its approximation capability}, volume={20}, ISSN={0749-159X 1098-2426}, url={http://dx.doi.org/10.1002/num.10092}, DOI={10.1002/num.10092}, abstractNote={This article discusses an immersed finite element (IFE) space introduced for solving a second-order elliptic boundary value problem with discontinuous coefficients (interface problem). The IFE space is nonconforming and its partition can be independent of the interface. The error estimates for the interpolation of a function in the usual Sobolev space indicate that this IFE space has an approximation capability similar to that of the standard conforming linear finite element space based on body-fit partitions. Numerical examples of the related finite element method based on this IFE space are provided. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 338–367, 2004}, number={3}, journal={Numerical Methods for Partial Differential Equations}, publisher={Wiley}, author={Li, Z. and Lin, T. and Lin, Y. and Rogers, R. C.}, year={2004}, pages={338–367} }
 @article{zolotarevskii_li_caraus_2004, title={Approximate solution of singular integro-differential equations by reduction over Faber-Laurent polynomials}, volume={40}, ISSN={["0012-2661"]}, DOI={10.1007/s10625-005-0108-3}, number={12}, journal={DIFFERENTIAL EQUATIONS}, author={Zolotarevskii, VA and Li, ZL and Caraus, I}, year={2004}, month={Dec}, pages={1764–1769} }
 @article{li_wang_2003, title={A Fast Finite Differenc Method For Solving Navier-Stokes Equations on Irregular Domains}, volume={1}, ISSN={1539-6746 1945-0796}, url={http://dx.doi.org/10.4310/cms.2003.v1.n1.a11}, DOI={10.4310/cms.2003.v1.n1.a11}, abstractNote={A fast finite difference method is proposed to solve the incompressible
Navier-Stokes equations defined on a general domain. The method is based 
on the voricity stream-function formulation and a fast Poisson solver defined 
on a general domain using the immersed interface method. The key to the new 
method is the fast Poisson solver for general domains and the interpolation 
scheme for the boundary condition of the stream function. Numerical examples
thats show second order accuracy of the computed solution are also provided.}, number={1}, journal={Communications in Mathematical Sciences}, publisher={International Press of Boston}, author={Li, Zhilin and Wang, Cheng}, year={2003}, pages={180–196} }
 @article{li_2003, title={An overview of the immersed interface method and its applications}, volume={7}, DOI={10.11650/twjm/1500407515}, abstractNote={Interface problems have many applications. Mathematically, interface problems usually lead to differential equations whose input data and solutions are non-smooth or discontinuous across some interfaces. The immersed interface method (IIM) has been developed in recent years particularly designed for interface problems. The IIM is a sharp interface method based on Cartesian grids. The IIM makes use of the jump conditions across the interface so that the finite difference/element discretization can be accurate. In this survey paper, we will introduce the immersed interface method for various problems, discuss its recent advances and related software packages, and some of its applications. We also review some other related methods and references in this survey paper.}, number={1}, journal={Taiwanese Journal of Mathematics}, author={Li, Zhilin}, year={2003}, pages={1–49} }
 @article{li_lin_torres_zhao_2003, title={Generalized Snell's Law for Weighted Minimal Surface in Heterogeneous Media}, volume={10}, ISSN={1073-2772 1945-0001}, url={http://dx.doi.org/10.4310/maa.2003.v10.n2.a3}, DOI={10.4310/maa.2003.v10.n2.a3}, abstractNote={The weighted minimal surface problem in heterogeneous media is studied in this paper. The solution to the weighted minimal surface problem is continuous but the derivatives have a jump across the interface where the medium property is discontinuous. The jump condition of the derivatives derived in this paper generalized the Snell's law in geometric optics to weighted minimal surfaces of co-dimension one in any dimensional space. A numerical method based on the gradient flow and the maximum principal preserving immersed interface method is developed to solve this nonlinear elliptic interface problem with jump conditions. Numerical computations are presented to verify both the analysis and the numerical algorithm. 1. Introduction. The minimal surface problem, that is, the problem of finding the surface of the least area among all surfaces having fixed boundary data, has been extensively studied. A recent workshop on minimal surfaces presented the latest re- search on minimal surface applications in chemistry and biology (5). Many phenomena that occur in nature relate to this problem which has been a motivation for devel- oping new mathematical theories and techniques to solve the problem analytically and numerically. Minimal surfaces were shown to be important in various chemical micro-structures and their corresponding phase transitions (5). Computer graphics and image analysis use minimal surfaces frequently for boundary detection, and to construct surfaces that are visually appealing (2), (15). Soap films and other mem- branes passing through a fixed boundary provide mechanical examples of minimal surfaces (14). A related concept is the idea of capillary surfaces, which result from surface tension in liquids. These surfaces are closely related to minimal surfaces (6). For a precise mathematical description of the minimal surface problem we refer, for example, to the classical treatises (7) and (16). The minimal surface problem can be described in two different ways, using the parametric or the non-parametric formulation. In the non-parametric setting the candidate surfaces are graphs of functions, while in the parametric setting the surfaces are treated as boundaries of sets (7). The former is usually seen in more physically- based treatments of the problem, whereas the later provides an excellent framework for the mathematical analysis of minimal surfaces. When the medium is homogeneous, the energy density at each point is constant, and therefore the surface energy is equivalent to the surface area. This is the standard minimal surface problem. In this paper, we consider the weighted minimal surface problem in a heterogeneous medium in which the energy density is piecewise smooth. For example this is the case for capillary interfaces in porous media or composite materials. In particular we derive a jump condition for the weighted minimal surface at the interface between two different media. The jump condition can be regarded as a generalized Snell's law which describes the refraction of minimal surfaces instead of light rays in geometric optics.}, number={2}, journal={Methods and Applications of Analysis}, publisher={International Press of Boston}, author={Li, Zhilin and Lin, Xiaobiao and Torres, Monica and Zhao, Hongkai}, year={2003}, pages={199–214} }
 @article{li_lin_wu_2003, title={New Cartesian grid methods for interface problems using the finite element formulation}, volume={96}, ISSN={["0029-599X"]}, DOI={10.1007/s00211-003-0473-x}, abstractNote={New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. The conforming finite element method has been shown to be convergent. With Cartesian triangulations, these new methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis.}, number={1}, journal={NUMERISCHE MATHEMATIK}, author={Li, ZL and Lin, T and Wu, XH}, year={2003}, month={Nov}, pages={61–98} }
 @article{li_wang_chern_lai_2003, title={New formulations for interface problems in polar coordinates}, volume={25}, ISSN={["1095-7197"]}, DOI={10.1137/S106482750139618X}, abstractNote={In this paper, numerical methods are proposed for some interface problems in polar or Cartesian coordinates. The new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution into a problem with a smooth solution. The new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution. In conjunction with the fast immersed interface method, a fast solver has been developed for the interface problems with a piecewise constant but a discontinuous coefficient using the new formulation in a polar coordinate system.}, number={1}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Li, ZL and Wang, WC and Chern, IL and Lai, MC}, year={2003}, pages={224–245} }
 @article{ito_li_2003, title={Solving a nonlinear problem in magneto-rheological fluids using the immersed interface method}, volume={19}, ISSN={["0885-7474"]}, DOI={10.1023/A:1025356025745}, number={1-3}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Ito, K and Li, ZL}, year={2003}, month={Dec}, pages={253–266} }
 @article{yang_li_li_2003, title={The  immersed interface method for elasticity problems with interfaces}, volume={10}, number={5}, journal={Dynamics of Continuous, Discrete & Impulsive Systems. Series A, Mathematical Analysis}, author={Yang, X. Z. and Li, B. and Li, Z. L.}, year={2003}, pages={783–808} }
 @article{deng_ito_li_2003, title={Three-dimensional elliptic solvers for interface problems and applications}, volume={184}, ISSN={["0021-9991"]}, DOI={10.1016/S0021-9991(02)00028-1}, abstractNote={Second-order accurate elliptic solvers using Cartesian grids are presented for three-dimensional interface problems in which the coefficients, the source term, the solution and its normal flux may be discontinuous across an interface. One of our methods is designed for general interface problems with variable but discontinuous coefficient. The scheme preserves the discrete maximum principle using constrained optimization techniques. An algebraic multigrid solver is applied to solve the discrete system. The second method is designed for interface problems with piecewise constant coefficient. The method is based on the fast immersed interface method and a fast 3D Poisson solver. The second method has been modified to solve Helmholtz/Poisson equations on irregular domains. An application of our method to an inverse interface problem of shape identification is also presented. In this application, the level set method is applied to find the unknown surface iteratively.}, number={1}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Deng, SZ and Ito, K and Li, ZL}, year={2003}, month={Jan}, pages={215–243} }
 @article{lubkin_li_2002, title={Force and deformation on branching rudiments: cleaving between hypotheses}, volume={1}, ISSN={["1617-7959"]}, url={http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=ORCID&SrcApp=OrcidOrg&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=WOS:000208282600003&KeyUID=WOS:000208282600003}, DOI={10.1007/s10237-002-0001-4}, number={1}, journal={BIOMECHANICS AND MODELING IN MECHANOBIOLOGY}, author={Lubkin, S. R. and Li, Z.}, year={2002}, month={Jun}, pages={5–16} }
 @book{gremaud_li_smith_tran_2002, title={Industrial Mathematics Modeling Workshop for Graduate Students Series}, year={2002} }
 @article{hunter_li_zhao_2002, title={Reactive autophobic spreading of drops}, volume={183}, ISSN={["1090-2716"]}, DOI={10.1006/jcph.2002.7168}, abstractNote={We use a lubrication theory approximation to formulate a model for the reactive spreading of drops that deposit an autophobic monolayer of surfactant on a surface. The model consists of a Poisson equation on a moving domain with boundary conditions that depend on the history of the domain motion. We develop a numerical algorithm for solving the model, using the immersed interface method and the level-set method. Numerical solutions for traveling drops are qualitatively similar to experimental observations of reactive autophobic spreading.}, number={2}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Hunter, JK and Li, ZL and Zhao, HK}, year={2002}, month={Dec}, pages={335–366} }
 @article{adams_li_2002, title={The immersed interface/multigrid methods for interface problems}, volume={24}, ISSN={["1064-8275"]}, DOI={10.1137/S1064827501389849}, abstractNote={New multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic PDEs that involve interfaces and discontinuities. For elliptic interface problems, the multigrid solver developed in this paper works while some other multigrid solvers do not. For linear parabolic equations, we have developed the second order maximum principle preserving finite difference scheme in this paper. We use the Crank--Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives. Numerical examples are also presented.}, number={2}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Adams, L and Li, ZL}, year={2002}, month={Oct}, pages={463–479} }
 @inbook{li_cai_2001, title={A Level Set-Boundary Element Method for Simulation of Dynamic Powder Consolidation of Metals}, volume={1988}, ISBN={9783540418146 9783540452621}, ISSN={0302-9743}, url={http://dx.doi.org/10.1007/3-540-45262-1_62}, DOI={10.1007/3-540-45262-1_62}, booktitle={Lecture Notes in Computer Science}, publisher={Springer Berlin Heidelberg}, author={Li, Zhilin and Cai, Wei}, editor={L. Vulkov, J. Wasniewski and Yalamov, P.Editors}, year={2001}, pages={527–534} }
 @article{lai_li_2001, title={A remark on jump conditions for the three-dimensional Navier-Stokes equations involving an immersed moving membrane}, volume={14}, ISSN={["0893-9659"]}, DOI={10.1016/S0893-9659(00)00127-0}, abstractNote={Jump conditions for the pressure, the velocity, and their normal derivatives across an immersed moving membrane in an incompressible fluid are derived. The discontinuities are due to the singular forces along the membrane. Instead of using the delta function formulation, those jump conditions can be used to formulate the governing equations in an alternative form. It is also useful for developing more accurate numerical methods such as immersed interface method for the Navier-Stokes equations involving moving interface.}, number={2}, journal={APPLIED MATHEMATICS LETTERS}, author={Lai, MC and Li, ZL}, year={2001}, month={Feb}, pages={149–154} }
 @misc{li_2001, title={Book Review: Generalized Difference Methods for Differential Equations}, volume={43}, number={1}, journal={SIAM Review}, author={Li, Zhilin}, year={2001}, pages={203–205,} }
 @book{gremaud_li_smith_tran_2001, title={Industrial Mathematics Modeling Workshop for Graduate Students Series}, year={2001} }
 @article{ito_kunisch_li_2001, title={Level-set function approach to an inverse interface problem}, volume={17}, ISSN={["1361-6420"]}, DOI={10.1088/0266-5611/17/5/301}, abstractNote={A model problem in electrical impedance tomography for the identification of unknown shapes from data in a narrow strip along the boundary of the domain is investigated. The representation of the shape of the boundary and its evolution during an iterative reconstruction process is achieved by the level set method. The shape derivatives of this problem involve the normal derivative of the potential along the unknown boundary. Hence an accurate resolution of its derivatives along the unknown interface is essential. It is obtained by the immersed interface method.}, number={5}, journal={INVERSE PROBLEMS}, author={Ito, K and Kunisch, K and Li, ZL}, year={2001}, month={Oct}, pages={1225–1242} }
 @article{li_ito_2001, title={Maximum principle preserving schemes for interface problems with discontinuous coefficients}, volume={23}, ISSN={["1064-8275"]}, DOI={10.1137/S1064827500370160}, abstractNote={New finite difference methods using Cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients, singular sources, and nonsmooth or even discontinuous solutions. The new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques. The methods are shown to converge under certain conditions using comparison functions. The coefficient matrix of the resulting linear system of equations is an M-matrix and is coupled with a multigrid solver. Numerical examples are also provided to show the efficiency of the proposed methods.}, number={1}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Li, ZL and Ito, K}, year={2001}, month={Jun}, pages={339–361} }
 @inbook{li_2001, place={Beijing}, title={Numerical Method for Simulation of Bubbles Flowing Through Another Fluid}, ISBN={9787030094452}, booktitle={Advances in Scientific Computing}, publisher={Science Pr}, author={Li, Zhilin}, editor={Mu, M. and Shi, Z. and Xue, W. and Zou, J.Editors}, year={2001}, pages={74–81} }
 @article{li_lubkin_2001, title={Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension}, volume={37}, ISSN={["0271-2091"]}, url={http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=ORCID&SrcApp=OrcidOrg&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=WOS:000171932200002&KeyUID=WOS:000171932200002}, DOI={10.1002/fld.185}, abstractNote={A fluid model of the incompressible Stokes equations in two space dimensions is used to simulate the motion of a droplet boundary separating two fluids with unequal viscosity and variable surface tension. Our theoretical analysis leads to decoupled jump conditions that are used in constructing the numerical algorithm. Numerical results agree with others in the literature and include some new findings that may apply to processes similar to cell cleavage. The method developed here accurately preserves area for our test problems. Some interesting observations are obtained with different choices of the parameters. Copyright © 2001 John Wiley & Sons, Ltd.}, number={5}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Li, ZL and Lubkin, SR}, year={2001}, month={Nov}, pages={525–540} }
 @article{li_lai_2001, title={The immersed interface method for the Navier-Stokes equations with singular forces}, volume={171}, ISSN={["1090-2716"]}, DOI={10.1006/jcph.2001.6813}, abstractNote={Abstract Peskin's Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be first-order accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a second-order projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully second-order accuracy for the velocity and nearly second-order accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper.}, number={2}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Li, ZL and Lai, MC}, year={2001}, month={Aug}, pages={822–842} }
 @book{gremaud_li_smith_tran_2000, place={Philadelphia}, title={Industrial Mathematics}, ISBN={O-89871-467-2}, publisher={SIAM}, year={2000} }
 @book{gremaud_li_smith_tran_2000, title={Industrial Mathematics Modeling Workshop for Graduate Students Series}, year={2000} }
 @article{li_shen_1999, title={A numerical method for solving heat equations involving interfaces}, volume={3}, journal={Electronic Journal of Differential Equations}, author={Li, Zhilin and Shen, Yun-Qui}, year={1999}, pages={100–108} }
 @article{li_zhao_gao_1999, title={A numerical study of electro-migration voiding by evolving level set functions on a fixed Cartesian grid}, volume={152}, ISSN={["0021-9991"]}, DOI={10.1006/jcph.1999.6249}, abstractNote={A numerical method for studying migration of voids driven by surface diffusion and electric current in a metal conducting line is developed. The mathematical model involves moving boundaries governed by a fourth order nonlinear partial differential equation which contains a nonlocal term corresponding to the electrical field and a nonlinear term corresponding to the curvature. Numerical challenges include efficient computation of the electrical field with sufficient accuracy to afford fourth order differentiation along the void boundary and to capture singularities arising in topological changes. We use the modified immersed interface method with a fixed Cartesian grid to solve for the electrical field, and the fast local level set method to update the position of moving voids. Numerical examples are performed to demonstrate the physical mechanisms by which voids interact under electromigration.}, number={1}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Li, ZL and Zhao, HK and Gao, HJ}, year={1999}, month={Jun}, pages={281–304} }
 @article{huang_li_1999, title={Convergence analysis of the immersed interface method}, volume={19}, ISSN={["0272-4979"]}, DOI={10.1093/imanum/19.4.583}, abstractNote={A rigorous error analysis is given for the immersed interface method (IIM) applied to elliptic problems with discontinuities and singularities. The finite difference scheme using IIM is shown to satisfy the conditions of a maximum principle for a certain class of problems. Comparison functions are constructed to obtain error bounds for some of the approximate solutions. The asymptotic error expansion provides further useful insights and details of the behaviour and convergence properties of IIM, which leads to a sharper estimate of the error bound. Second-order convergence of IIM is indicated by the analysis. Numerical examples are also given to support the analytical results.}, number={4}, journal={IMA JOURNAL OF NUMERICAL ANALYSIS}, author={Huang, HX and Li, ZL}, year={1999}, month={Oct}, pages={583–608} }
 @article{wiegmann_li_leveque_1999, title={Crack jump conditions for elliptic problems}, volume={12}, ISSN={["0893-9659"]}, DOI={10.1016/S0893-9659(99)00083-X}, abstractNote={We derive jump conditions for a potential function and its derivatives across a crack. A crack is a “thin” region of very different conductivity, for example a fracture in otherwise homogeneous material. Such a sharp change of material properties introduces a discontinuity in the coefficient of the elliptic equation governing the potential. The crack cannot be neglected, because it substantially alters the behavior of the potential. Numerically, it is very difficult to resolve the potential near the crack. A strategy is to treat the crack as a lower dimensional interface (hypersurface). Jump conditions across the crack for the potential and its derivatives are necessary for the development of numerical schemes for this approach. Besides the jump conditions, we also give an analytic example of their validity.}, number={6}, journal={APPLIED MATHEMATICS LETTERS}, author={Wiegmann, A and Li, Z and LeVeque, RJ}, year={1999}, month={Aug}, pages={81–88} }
 @article{li_soni_1999, title={Fast and accurate numerical approaches for Stefan problems and crystal growth}, volume={35}, ISSN={["1040-7790"]}, DOI={10.1080/104077999275848}, abstractNote={New numerical approaches for moving boundary interface applications tailored for Stefan problems and crystal growth simulation are proposed in this article. The focus is on the issues of accuracy and speed-up. A modified Crank-Nicolson method that is second-order accurate and stable is developed. The alternating directional implicit (ADI) method is also developed to speed up the simulation for a certain class of problems. The ADI method is shown to be asymptotically stable and at least first-order accurate. Numerical results, however, show that the ADI method actually provides second-order accuracy if the velocity can be calculated accurately. The level set method is used to update the moving interface so that the topological changes can be handled easily. Numerical experiments are compared to exact solutions and results in the literature.}, number={4}, journal={NUMERICAL HEAT TRANSFER PART B-FUNDAMENTALS}, author={Li, ZL and Soni, B}, year={1999}, month={Jun}, pages={461–484} }
 @book{gremaud_li_smith_tran_1999, title={Industrial Mathematics Modeling Workshop for Graduate Students series}, year={1999} }
 @article{gao_li_zhao_1999, title={Numerical Study of Two Dimensional Electro-migration}, volume={152}, journal={Journal of Computational Physics}, author={Gao, H and Li, Zhilin and Zhao, H}, year={1999}, pages={281–304} }
 @article{ewing_li_lin_lin_1999, title={The immersed finite volume element methods for the elliptic interface problems}, volume={50}, ISSN={["0378-4754"]}, DOI={10.1016/S0378-4754(99)00061-0}, abstractNote={An immersed finite element space is used to solve the elliptic interface problems by a finite volume element method. Special nodal basis functions are introduced in a triangle whose interior intersects with the interface so that the jump conditions across the interface are satisfied. Optimal error estimates in an energy norm are obtained. Numerical results are supplied to justify the theoretical work and to reveal some interesting features of the method.}, number={1-4}, journal={MATHEMATICS AND COMPUTERS IN SIMULATION}, author={Ewing, RE and Li, ZL and Lin, T and Lin, YP}, year={1999}, month={Nov}, pages={63–76} }
 @article{li_putcha_1999, title={Types of reductive monoids}, volume={221}, DOI={10.1006/jabr.1999.7946}, abstractNote={Let M be a reductive monoid with a reductive unit group G. Clearly there is a natural G × G action on M. The orbits are the J-classes (in the sense of semigroup theory) and form a finite lattice. The general problem of finding the lattice remains open. In this paper we study a new class of reductive monoids constructed by multilined closure. We obtain a general theorem to determine the lattices of these monoids. We find that the (J, σ)-irreducible monoids of Suzuki type and Ree type belong to this new class. Using the general theorem we then list all the lattices and type maps of the (J, σ)-irreducible monoids of Suzuki type and Ree type.}, number={1}, journal={Journal of Algebra}, author={Li, Zhilin and Putcha, M.}, year={1999}, pages={102–116} }
 @article{li_1998, title={A fast iterative algorithm for elliptic interface problems}, volume={35}, ISSN={["1095-7170"]}, DOI={10.1137/S0036142995291329}, abstractNote={A fast, second-order accurate iterative method is proposed for the elliptic equation \[ \grad\cdot(\beta(x,y) \grad u) =f(x,y) \] in a rectangularregion $\Omega$ in two-space dimensions. We assume that there is an irregular interface across which the coefficient $\beta$, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the coefficients $\beta$ are piecewise constant and the jump in $\beta$ is large. The interface may or may not align with an underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [ SIAM J. Numer. Anal., 4 (1994), pp. 1019--1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second-order difference scheme for a corresponding Poisson equation in the region, and a second-order discretization for a Neumann-like interface condition. Thus second-order accuracy is guaranteed. A GMRES iteration is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is also proposed to approximate interface quantities from a grid function. Numerical experiments are provided and analyzed. The number of iterations in solving the Schur complement system appears to be independent of both the jump in the coefficient and the mesh size.}, number={1}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Li, ZL}, year={1998}, month={Feb}, pages={230–254} }
 @article{li_1998, title={The immersed interface method using a finite element formulation}, volume={27}, ISSN={["0168-9274"]}, DOI={10.1016/S0168-9274(98)00015-4}, abstractNote={A finite element method is proposed for one dimensional interface problems involving discontinuities in the coefficients of the differential equations and the derivatives of the solutions. The interfaces do not have to be one of grid points. The idea is to construct basis functions which satisfy the interface jump conditions. By constructing an interpolating function of the solution, we are able to give a rigorous error analysis which shows that the approximate solution obtained from the finite element method is second order accurate in the infinity norm. Numerical examples are also provided to support the method and the theoretical analysis. Several numerical approaches are also proposed for dealing with two dimensional problems involving interfaces.}, number={3}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Li, ZL}, year={1998}, month={Jul}, pages={253–267} }
 @article{li_wang_zou_1998, title={Theoretical and numerical analysis on a thermo-elastic system with discontinuities}, volume={92}, ISSN={0377-0427}, url={http://dx.doi.org/10.1016/s0377-0427(98)00044-2}, DOI={10.1016/s0377-0427(98)00044-2}, abstractNote={A second-order accurate numerical scheme is proposed for a thermo-elastic system which models a bar made of two distinct materials. The physical parameters involved may be discontinuous across the joint of the two materials, where there might be also singular heat and/or force sources. The solution components, the temperature and the displacement, may change rapidly across the joint. By transforming the system into a different one, time-marching schemes can be used for the new system which is well posed. The immersed interface method is employed to deal with the discontinuities of the coefficients and the singular sources. The proposed numerical method can fit both explicit and implicit formulation. For the implicit version, a stable and fast prediction-correction scheme is also developed. Convergence analysis shows that our method is second-order accurate at all grid points in spite of the discontinuities across the interface. Numerical experiments are performed to support the theoretical analysis in this paper.}, number={1}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier BV}, author={Li, Zhilin and Wang, Deshen and Zou, Jun}, year={1998}, month={May}, pages={37–58} }
 @article{hou_li_osher_zhao_1997, title={A Hybrid Method for Moving Interface Problems with Application to the Hele–Shaw Flow}, volume={134}, ISSN={0021-9991}, url={http://dx.doi.org/10.1006/jcph.1997.5689}, DOI={10.1006/jcph.1997.5689}, abstractNote={In this paper, a hybrid approach which combines theimmersed interface methodwith thelevel set approachis presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to the discontinuity of the coefficients or/and singular sources along the interfaces. The moving interfaces then are updated using the newly developed fast level set formulation which involves computation only inside some small tubes containing the interfaces. This method combines the advantage of the two approaches and gives a second-order Eulerian discretization for interface problems. Several key steps in the implementation are addressed in detail. This new approach is then applied to Hele?Shaw flow, an unstable flow involving two fluids with very different viscosity.}, number={2}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Hou, Thomas Y. and Li, Zhilin and Osher, Stanley and Zhao, Hongkai}, year={1997}, month={Jul}, pages={236–252} }
 @article{li_zheng_1997, title={An Inverse Problem in a Parabolic Equation}, volume={1}, journal={Electronic Journal of Differential Equations}, author={Li, Zhilin and Zheng, K}, year={1997}, pages={193–199} }
 @article{heine_li_mctigue_1997, title={Front fixing vs. front tracking for diffusive transport with moving boundaries}, volume={21}, journal={International Journal for Numerical & Analytical Methods in Geomechanics}, author={Heine, J. T. and Li, Zhilin and McTigue, D. F.}, year={1997}, pages={653–662} }
 @article{leveque_li_1997, title={Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension}, volume={18}, ISSN={1064-8275 1095-7197}, url={http://dx.doi.org/10.1137/s1064827595282532}, DOI={10.1137/s1064827595282532}, abstractNote={A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second-order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019--1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.}, number={3}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={LeVeque, Randall J. and Li, Zhilin}, year={1997}, month={May}, pages={709–735} }
 @article{li_1997, title={Immersed interface methods for moving interface problems}, volume={14}, DOI={10.1023/A:1019173215885}, journal={Numerical Algorithms}, author={Li, Zhilin}, year={1997}, month={May}, pages={269–293} }
 @article{li_1996, title={A note on immersed interface method for three-dimensional elliptic equations}, volume={31}, ISSN={0898-1221}, url={http://dx.doi.org/10.1016/0898-1221(95)00202-2}, DOI={10.1016/0898-1221(95)00202-2}, abstractNote={The Immersed Interface Method proposed by LeVeque and Li [1] is extended to three-dimensional elliptic equations of the form: ∇·(β(x)∇u(x))+κ(x)u(x)=f(x). We study the situation in which there is an irregular interface (surface) S contained in the solution domain across which β, κ and f may be discontinuous or even singular. As a result, the solution u will usually be nonsmooth or even discontinuous. A finite difference approach with a uniform Cartesian grid is used in the discretization. Local truncation error analysis is performed to estimate the accuracy of the numerical solution.}, number={3}, journal={Computers & Mathematics with Applications}, publisher={Elsevier BV}, author={Li, Z.}, year={1996}, month={Feb}, pages={9–17} }
 @inproceedings{leveque_li_1995, title={Simulation of bubbles in creeping flow using the immersed interface method}, booktitle={Proceedings of the sixth international symposium on computational fluid dynamics}, author={LeVeque, R. J. and Li, Zhilin}, year={1995}, pages={688–693} }
 @article{leveque_li_1994, title={The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources}, volume={31}, ISSN={0036-1429 1095-7170}, url={http://dx.doi.org/10.1137/0731054}, DOI={10.1137/0731054}, abstractNote={The authors develop finite difference methods for elliptic equations of the form \[ \nabla \cdot (\beta (x)\nabla u(x)) + \kappa (x)u(x) = f(x)\] in a region $\Omega $ in one or two space dimensions. It is assumed that $\Omega $ is a simple region (e.g., a rectangle) and that a uniform rectangular grid is used. The situation is studied in which there is an irregular surface $\Gamma $ of codimension 1 contained in $\Omega $ across which $\beta ,\kappa $, and f may be discontinuous, and along which the source f may have a delta function singularity. As a result, derivatives of the solution u may be discontinuous across $\Gamma $. The specification of a jump discontinuity in u itself across $\Gamma $ is allowed. It is shown that it is possible to modify the standard centered difference approximation to maintain second order accuracy on the uniform grid even when $\Gamma $ is not aligned with the grid. This approach is also compared with a discrete delta function approach to handling singular sources, as used in Peskin’s immersed boundary method.}, number={4}, journal={SIAM Journal on Numerical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={LeVeque, Randall J. and Li, Zhilin}, year={1994}, month={Aug}, pages={1019–1044} }
 @phdthesis{li_1994, title={The Immersed Interface Method — A Numerical Approach for Partial Differential Equations with Interfaces}, school={University of Washington}, author={Li, Zhilin}, year={1994} }
 @inproceedings{li_mayo_1993, place={Providence, Rhode Island}, title={ADI methods for heat equations with discontinuities along an arbitrary interface}, volume={48}, booktitle={Proceedings of Symposia in Applied Mathematics}, publisher={AMS}, author={Li, Zhilin and Mayo, A}, editor={Gautschi, W.Editor}, year={1993}, pages={311–315} }
 @article{huang_li_1990, title={Perturbation theorems of eigenvectors}, volume={12}, number={3}, journal={Numerical Mathematics, a Journal of Chinese Universities}, author={Huang, K and Li, Zhilin}, year={1990}, pages={284–289} }
 @article{huang_li_1989, series={with K}, title={Roundoff error analysis for polynomial interpolation}, number={2}, journal={Research and Review in Mathematics}, publisher={Huang}, author={Huang, K and Li, Zhilin}, year={1989}, collection={with K} }
 @article{li_1989, title={The uniform treatment for linear system — Algorithm and numerical stability}, volume={10}, number={2}, journal={Journal on Numerical Methods & Computer Applications}, author={Li, Zhilin}, year={1989} }
 @article{li_1988, title={Optimal conjugate gradient method for solving arbitrary linear equations}, number={3}, journal={Journal of Nanjing Normal University (Natural Science Edition)}, author={Li, Zhilin}, year={1988}, pages={28–34} }
 @article{li_1987, title={A generalized conjugate gradient method for solving real skew-symmetric systems}, volume={8}, number={4}, journal={Journal on Numerical Methods & Computer Applications}, author={Li, Zhilin}, year={1987}, pages={31–37} }
 @article{huang_li_1987, title={An equivalent theorem on the numerical stability for an algorithm}, volume={9}, number={1}, journal={Numerical Mathematics, a Journal of Chinese Universities}, author={Huang, K and Li, Zhilin}, year={1987}, pages={59–65} }
 @article{huang_li_1985, title={On the relation between the behavior and the distribution of the zeros of a polynomial}, volume={2}, number={1}, journal={Journal of Nanjing University, Mathematics Biquarterly}, author={Huang, K and Li, Zhilin}, year={1985}, pages={53–59} }
 @book{industrial mathematics, journal={SIAM, USA} }
 @book{numerical analysis and its applications, url={https://link.springer.com/book/10.1007%2Fb106395} }