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Volume 6, Issue 4
Convergent Finite Elements on Arbitrary Meshes, the WG Method

Ran Zhang & Shangyou Zhang

DOI: 10.4208/csiam-am.SO-2024-0038

CSIAM Trans. Appl. Math., 6 (2025), pp. 651-665.

Published online: 2025-09

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  • Abstract

On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e. on arbitrary meshes. Numerical tests confirm the theory.

  • Keywords

Discontinuous finite element, maximum angle condition, Poisson’s equation, triangular grid, tetrahedral grid.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-6-651, author = {Zhang , Ran and Zhang , Shangyou}, title = {Convergent Finite Elements on Arbitrary Meshes, the WG Method}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2025}, volume = {6}, number = {4}, pages = {651--665}, abstract = {

On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e. on arbitrary meshes. Numerical tests confirm the theory.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2024-0038}, url = {http://global-sci.org/intro/article_detail/csiam-am/24499.html} }
TY - JOUR T1 - Convergent Finite Elements on Arbitrary Meshes, the WG Method AU - Zhang , Ran AU - Zhang , Shangyou JO - CSIAM Transactions on Applied Mathematics VL - 4 SP - 651 EP - 665 PY - 2025 DA - 2025/09 SN - 6 DO - http://doi.org/10.4208/csiam-am.SO-2024-0038 UR - https://global-sci.org/intro/article_detail/csiam-am/24499.html KW - Discontinuous finite element, maximum angle condition, Poisson’s equation, triangular grid, tetrahedral grid. AB -

On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e. on arbitrary meshes. Numerical tests confirm the theory.

Zhang , Ran and Zhang , Shangyou. (2025). Convergent Finite Elements on Arbitrary Meshes, the WG Method. CSIAM Transactions on Applied Mathematics. 6 (4). 651-665. doi:10.4208/csiam-am.SO-2024-0038
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