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Volume 20, Issue 6
A Conforming DG Method for the Biharmonic Equation on Polytopal Meshes

Xiu Ye & Shangyou Zhang

DOI: 10.4208/ijnam2023-1037

Int. J. Numer. Anal. Mod., 20 (2023), pp. 855-869.

Published online: 2023-11

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  • Abstract

A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.

  • Keywords

finite element methods, weak Laplacian, biharmonic equations, polyhedral meshes.

  • AMS Subject Headings

65N15, 65N30, 76D07

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{IJNAM-20-855, author = {Ye , Xiu and Zhang , Shangyou}, title = {A Conforming DG Method for the Biharmonic Equation on Polytopal Meshes}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {6}, pages = {855--869}, abstract = {

A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1037}, url = {http://global-sci.org/intro/article_detail/ijnam/22144.html} }
TY - JOUR T1 - A Conforming DG Method for the Biharmonic Equation on Polytopal Meshes AU - Ye , Xiu AU - Zhang , Shangyou JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 855 EP - 869 PY - 2023 DA - 2023/11 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1037 UR - https://global-sci.org/intro/article_detail/ijnam/22144.html KW - finite element methods, weak Laplacian, biharmonic equations, polyhedral meshes. AB -

A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.

Xiu Ye & Shangyou Zhang. (2023). A Conforming DG Method for the Biharmonic Equation on Polytopal Meshes. International Journal of Numerical Analysis and Modeling. 20 (6). 855-869. doi:10.4208/ijnam2023-1037
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