@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{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{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{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 not only to get a second order accurate solution but also a second order accurate gradient from each side of the interface. The key of the new method is to introduce the jump in the normal derivative of the solution as an augmented variable and re-write 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 derivatives 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 near or on the interface so that the resulting coefficient matrix is an M-matrix. A multi-grid 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 has also been proved in this paper. Numerical examples for general elliptic interface problems have confirmed 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} }