Volume 40, Issue 6
An $L^{\infty}$ Second Order Cartesian Method for 3D Anisotropic Interface Problems

J. Comp. Math., 40 (2022), pp. 882-912.

Published online: 2022-08

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• Abstract

A second order accurate method in the infinity norm is proposed for general three dimensional anisotropic elliptic interface problems in which the solution and its derivatives, the coefficients, and source terms all can have finite jumps across one or several arbitrary smooth interfaces. The method is based on the 2D finite element-finite difference (FE-FD) method but with substantial differences in method derivation, implementation, and convergence analysis. One of challenges is to derive 3D interface relations since there is no invariance anymore under coordinate system transforms for the partial differential equations and the jump conditions. A finite element discretization whose coefficient matrix is a symmetric semi-positive definite is used away from the interface; and the maximum preserving finite difference discretization whose coefficient matrix part is an M-matrix is constructed at irregular elements where the interface cuts through. We aim to get a sharp interface method that can have second order accuracy in the point-wise norm. We show the convergence analysis by splitting errors into several parts. Nontrivial numerical examples are presented to confirm the convergence analysis.

65M06, 76M20, 65N06

baiying_dong@nxnu.edu.cn (Baiying Dong)

xf_feng@nxu.edu.cn (Xiufeng Feng)

zhilin@ncsu.edu (Zhilin Li)

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@Article{JCM-40-882, author = {Dong , BaiyingFeng , Xiufeng and Li , Zhilin}, title = {An $L^{\infty}$ Second Order Cartesian Method for 3D Anisotropic Interface Problems}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {6}, pages = {882--912}, abstract = {

A second order accurate method in the infinity norm is proposed for general three dimensional anisotropic elliptic interface problems in which the solution and its derivatives, the coefficients, and source terms all can have finite jumps across one or several arbitrary smooth interfaces. The method is based on the 2D finite element-finite difference (FE-FD) method but with substantial differences in method derivation, implementation, and convergence analysis. One of challenges is to derive 3D interface relations since there is no invariance anymore under coordinate system transforms for the partial differential equations and the jump conditions. A finite element discretization whose coefficient matrix is a symmetric semi-positive definite is used away from the interface; and the maximum preserving finite difference discretization whose coefficient matrix part is an M-matrix is constructed at irregular elements where the interface cuts through. We aim to get a sharp interface method that can have second order accuracy in the point-wise norm. We show the convergence analysis by splitting errors into several parts. Nontrivial numerical examples are presented to confirm the convergence analysis.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2103-m2020-0107}, url = {http://global-sci.org/intro/article_detail/jcm/20840.html} }
TY - JOUR T1 - An $L^{\infty}$ Second Order Cartesian Method for 3D Anisotropic Interface Problems AU - Dong , Baiying AU - Feng , Xiufeng AU - Li , Zhilin JO - Journal of Computational Mathematics VL - 6 SP - 882 EP - 912 PY - 2022 DA - 2022/08 SN - 40 DO - http://doi.org/10.4208/jcm.2103-m2020-0107 UR - https://global-sci.org/intro/article_detail/jcm/20840.html KW - 3D anisotropic PDE, Cartesian meshes, Finite element method, Finite difference method, Maximum preserving IIM, Convergence analysis. AB -

A second order accurate method in the infinity norm is proposed for general three dimensional anisotropic elliptic interface problems in which the solution and its derivatives, the coefficients, and source terms all can have finite jumps across one or several arbitrary smooth interfaces. The method is based on the 2D finite element-finite difference (FE-FD) method but with substantial differences in method derivation, implementation, and convergence analysis. One of challenges is to derive 3D interface relations since there is no invariance anymore under coordinate system transforms for the partial differential equations and the jump conditions. A finite element discretization whose coefficient matrix is a symmetric semi-positive definite is used away from the interface; and the maximum preserving finite difference discretization whose coefficient matrix part is an M-matrix is constructed at irregular elements where the interface cuts through. We aim to get a sharp interface method that can have second order accuracy in the point-wise norm. We show the convergence analysis by splitting errors into several parts. Nontrivial numerical examples are presented to confirm the convergence analysis.

Baiying Dong, Xiufeng Feng & Zhilin Li. (2022). An $L^{\infty}$ Second Order Cartesian Method for 3D Anisotropic Interface Problems. Journal of Computational Mathematics. 40 (6). 882-912. doi:10.4208/jcm.2103-m2020-0107
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