Volume 18, Issue 4
A Modified Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form

Int. J. Numer. Anal. Mod., 18 (2021), pp. 500-523.

Published online: 2021-05

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• Abstract

A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in [6], the system of equations resulting from the M-PDWG scheme could be equivalently simplified into one equation involving only the primal variable by eliminating the dual variable (Lagrange multiplier). The resulting simplified system thus has significantly fewer degrees of freedom than the one resulting from existing PDWG scheme. Optimal order error estimates are derived for the numerical approximations in the discrete $H^2$-norm, $H^1$-norm and $L^2$-norm respectively. Extensive numerical results are demonstrated for both the smooth and non-smooth coefficients on convex and non-convex domains to verify the accuracy of the theory developed in this paper.

• Keywords

Primal-dual, weak Galerkin, finite element methods, non-divergence form, Cordès condition, polyhedral meshes.

• BibTex
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@Article{IJNAM-18-500, author = {Wang , ​Chunmei}, title = {A Modified Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2021}, volume = {18}, number = {4}, pages = {500--523}, abstract = {

A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in [6], the system of equations resulting from the M-PDWG scheme could be equivalently simplified into one equation involving only the primal variable by eliminating the dual variable (Lagrange multiplier). The resulting simplified system thus has significantly fewer degrees of freedom than the one resulting from existing PDWG scheme. Optimal order error estimates are derived for the numerical approximations in the discrete $H^2$-norm, $H^1$-norm and $L^2$-norm respectively. Extensive numerical results are demonstrated for both the smooth and non-smooth coefficients on convex and non-convex domains to verify the accuracy of the theory developed in this paper.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/19112.html} }
TY - JOUR T1 - A Modified Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form AU - Wang , ​Chunmei JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 500 EP - 523 PY - 2021 DA - 2021/05 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/19112.html KW - Primal-dual, weak Galerkin, finite element methods, non-divergence form, Cordès condition, polyhedral meshes. AB -

A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in [6], the system of equations resulting from the M-PDWG scheme could be equivalently simplified into one equation involving only the primal variable by eliminating the dual variable (Lagrange multiplier). The resulting simplified system thus has significantly fewer degrees of freedom than the one resulting from existing PDWG scheme. Optimal order error estimates are derived for the numerical approximations in the discrete $H^2$-norm, $H^1$-norm and $L^2$-norm respectively. Extensive numerical results are demonstrated for both the smooth and non-smooth coefficients on convex and non-convex domains to verify the accuracy of the theory developed in this paper.

​Chunmei Wang. (2021). A Modified Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form. International Journal of Numerical Analysis and Modeling. 18 (4). 500-523. doi:
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