Volume 11, Issue 3
A Weak Galerkin Finite Element Method for Elliptic Interface Problems with Polynomial Reduction

Bhupen Deka

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 655-672.

Published online: 2018-11

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  • Abstract

This paper is concerned with numerical approximation of elliptic interface problems via week Galerkin (WG) finite element method. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme are significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimizes the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both $H^1$ and $L^2$ norms are established for the present WG finite element solutions.

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@Article{NMTMA-11-655, author = {}, title = {A Weak Galerkin Finite Element Method for Elliptic Interface Problems with Polynomial Reduction }, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {3}, pages = {655--672}, abstract = {

This paper is concerned with numerical approximation of elliptic interface problems via week Galerkin (WG) finite element method. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme are significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimizes the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both $H^1$ and $L^2$ norms are established for the present WG finite element solutions.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017-OA-0078}, url = {http://global-sci.org/intro/article_detail/nmtma/12448.html} }
TY - JOUR T1 - A Weak Galerkin Finite Element Method for Elliptic Interface Problems with Polynomial Reduction JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 655 EP - 672 PY - 2018 DA - 2018/11 SN - 11 DO - http://doi.org/10.4208/nmtma.2017-OA-0078 UR - https://global-sci.org/intro/article_detail/nmtma/12448.html KW - AB -

This paper is concerned with numerical approximation of elliptic interface problems via week Galerkin (WG) finite element method. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme are significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimizes the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both $H^1$ and $L^2$ norms are established for the present WG finite element solutions.

Bhupen Deka. (2020). A Weak Galerkin Finite Element Method for Elliptic Interface Problems with Polynomial Reduction . Numerical Mathematics: Theory, Methods and Applications. 11 (3). 655-672. doi:10.4208/nmtma.2017-OA-0078
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