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Volume 18, Issue 1
A Divergence-Free $P_k$ CDG Finite Element for the Stokes Equations on Triangular and Tetrahedral Meshes

Xiu Ye & Shangyou Zhang

DOI: 10.4208/nmtma.OA-2024-0063

Numer. Math. Theor. Meth. Appl., 18 (2025), pp. 157-174.

Published online: 2025-04

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  • Abstract

In the conforming discontinuous Galerkin method, the standard bilinear form for the conforming finite elements is applied to discontinuous finite elements without adding any inter-element nor penalty form. The $P_k$ $(k ≥ 1)$ discontinuous finite elements and the $P_{k−1}$ weak Galerkin finite elements are adopted to approximate the velocity and the pressure respectively, when solving the Stokes equations on triangular or tetrahedral meshes. The discontinuous finite element solutions are divergence-free and surprisingly H-div functions on the whole domain. The optimal order convergence is achieved for both variables and for all $k ≥ 1.$ The theory is verified by numerical examples.

  • Keywords

Finite element, conforming discontinuous Galerkin method, Stokes equations, stabilizer free, divergence free.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-18-157, author = {Ye , Xiu and Zhang , Shangyou}, title = {A Divergence-Free $P_k$ CDG Finite Element for the Stokes Equations on Triangular and Tetrahedral Meshes}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2025}, volume = {18}, number = {1}, pages = {157--174}, abstract = {

In the conforming discontinuous Galerkin method, the standard bilinear form for the conforming finite elements is applied to discontinuous finite elements without adding any inter-element nor penalty form. The $P_k$ $(k ≥ 1)$ discontinuous finite elements and the $P_{k−1}$ weak Galerkin finite elements are adopted to approximate the velocity and the pressure respectively, when solving the Stokes equations on triangular or tetrahedral meshes. The discontinuous finite element solutions are divergence-free and surprisingly H-div functions on the whole domain. The optimal order convergence is achieved for both variables and for all $k ≥ 1.$ The theory is verified by numerical examples.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0063}, url = {http://global-sci.org/intro/article_detail/nmtma/23945.html} }
TY - JOUR T1 - A Divergence-Free $P_k$ CDG Finite Element for the Stokes Equations on Triangular and Tetrahedral Meshes AU - Ye , Xiu AU - Zhang , Shangyou JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 157 EP - 174 PY - 2025 DA - 2025/04 SN - 18 DO - http://doi.org/10.4208/nmtma.OA-2024-0063 UR - https://global-sci.org/intro/article_detail/nmtma/23945.html KW - Finite element, conforming discontinuous Galerkin method, Stokes equations, stabilizer free, divergence free. AB -

In the conforming discontinuous Galerkin method, the standard bilinear form for the conforming finite elements is applied to discontinuous finite elements without adding any inter-element nor penalty form. The $P_k$ $(k ≥ 1)$ discontinuous finite elements and the $P_{k−1}$ weak Galerkin finite elements are adopted to approximate the velocity and the pressure respectively, when solving the Stokes equations on triangular or tetrahedral meshes. The discontinuous finite element solutions are divergence-free and surprisingly H-div functions on the whole domain. The optimal order convergence is achieved for both variables and for all $k ≥ 1.$ The theory is verified by numerical examples.

Ye , Xiu and Zhang , Shangyou. (2025). A Divergence-Free $P_k$ CDG Finite Element for the Stokes Equations on Triangular and Tetrahedral Meshes. Numerical Mathematics: Theory, Methods and Applications. 18 (1). 157-174. doi:10.4208/nmtma.OA-2024-0063
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