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Volume 20, Issue 5
A New Weak Galerkin Method with Weakly Enforced Dirichlet Boundary Condition

Dan Li, Yiqiang Li & Zhanbin Yuan

DOI: 10.4208/ijnam2023-1028

Int. J. Numer. Anal. Mod., 20 (2023), pp. 647-666.

Published online: 2023-09

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  • Abstract

A new weak Galerkin method with weakly enforced Dirichlet boundary condition is proposed and analyzed for the second order elliptic problems. Two penalty terms are incorporated into the weak Galerkin method to enforce the boundary condition in the weak sense. The new numerical scheme is designed by using the locally constructed weak gradient. Optimal order error estimates are established for the numerical approximation in the energy norm and usual $L^2$ norm. Moreover, by using the Schur complement technique, the unknowns of the numerical scheme are only defined on the boundary of each piecewise element and an effective implementation of the reduced global system is presented. Some numerical experiments are reported to demonstrate the accuracy and efficiency of the proposed method.

  • Keywords

weak Galerkin, finite element methods, weak gradient, second order elliptic problems, polytopal partitions.

  • AMS Subject Headings

35J50, 65N30, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-647, author = {Li , DanLi , Yiqiang and Yuan , Zhanbin}, title = {A New Weak Galerkin Method with Weakly Enforced Dirichlet Boundary Condition }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {5}, pages = {647--666}, abstract = {

A new weak Galerkin method with weakly enforced Dirichlet boundary condition is proposed and analyzed for the second order elliptic problems. Two penalty terms are incorporated into the weak Galerkin method to enforce the boundary condition in the weak sense. The new numerical scheme is designed by using the locally constructed weak gradient. Optimal order error estimates are established for the numerical approximation in the energy norm and usual $L^2$ norm. Moreover, by using the Schur complement technique, the unknowns of the numerical scheme are only defined on the boundary of each piecewise element and an effective implementation of the reduced global system is presented. Some numerical experiments are reported to demonstrate the accuracy and efficiency of the proposed method.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1028}, url = {http://global-sci.org/intro/article_detail/ijnam/22006.html} }
TY - JOUR T1 - A New Weak Galerkin Method with Weakly Enforced Dirichlet Boundary Condition AU - Li , Dan AU - Li , Yiqiang AU - Yuan , Zhanbin JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 647 EP - 666 PY - 2023 DA - 2023/09 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1028 UR - https://global-sci.org/intro/article_detail/ijnam/22006.html KW - weak Galerkin, finite element methods, weak gradient, second order elliptic problems, polytopal partitions. AB -

A new weak Galerkin method with weakly enforced Dirichlet boundary condition is proposed and analyzed for the second order elliptic problems. Two penalty terms are incorporated into the weak Galerkin method to enforce the boundary condition in the weak sense. The new numerical scheme is designed by using the locally constructed weak gradient. Optimal order error estimates are established for the numerical approximation in the energy norm and usual $L^2$ norm. Moreover, by using the Schur complement technique, the unknowns of the numerical scheme are only defined on the boundary of each piecewise element and an effective implementation of the reduced global system is presented. Some numerical experiments are reported to demonstrate the accuracy and efficiency of the proposed method.

Li , DanLi , Yiqiang and Yuan , Zhanbin. (2023). A New Weak Galerkin Method with Weakly Enforced Dirichlet Boundary Condition . International Journal of Numerical Analysis and Modeling. 20 (5). 647-666. doi:10.4208/ijnam2023-1028
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