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Volume 15, Issue 2
The Discontinuous Galerkin Method by Divergence-Free Patch Reconstruction for Stokes Eigenvalue Problems

Di Li, Zhiyuan Sun, Fengru Wang & Jerry Zhijian Yang

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 484-509.

Published online: 2022-03

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  • Abstract

The discontinuous Galerkin method by divergence-free patch reconstruction is proposed for Stokes eigenvalue problems. It utilizes the mixed finite element framework. The patch reconstruction technique constructs two categories of approximation spaces. Namely, the local divergence-free space is employed to discretize the velocity space, and the pressure space is approximated by standard reconstruction space simultaneously. Benefit from the divergence-free constraint; the identical element patch serves two approximation spaces while using the element pair $\mathbb{P}^{m+1}/ \mathbb{P}^m$. The optimal error estimate is derived under the inf-sup condition framework. Numerical examples are carried out to validate the inf-sup test and the theoretical results.

  • AMS Subject Headings

49N45, 65N21

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-484, author = {Li , DiSun , ZhiyuanWang , Fengru and Yang , Jerry Zhijian}, title = {The Discontinuous Galerkin Method by Divergence-Free Patch Reconstruction for Stokes Eigenvalue Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {2}, pages = {484--509}, abstract = {

The discontinuous Galerkin method by divergence-free patch reconstruction is proposed for Stokes eigenvalue problems. It utilizes the mixed finite element framework. The patch reconstruction technique constructs two categories of approximation spaces. Namely, the local divergence-free space is employed to discretize the velocity space, and the pressure space is approximated by standard reconstruction space simultaneously. Benefit from the divergence-free constraint; the identical element patch serves two approximation spaces while using the element pair $\mathbb{P}^{m+1}/ \mathbb{P}^m$. The optimal error estimate is derived under the inf-sup condition framework. Numerical examples are carried out to validate the inf-sup test and the theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0085}, url = {http://global-sci.org/intro/article_detail/nmtma/20361.html} }
TY - JOUR T1 - The Discontinuous Galerkin Method by Divergence-Free Patch Reconstruction for Stokes Eigenvalue Problems AU - Li , Di AU - Sun , Zhiyuan AU - Wang , Fengru AU - Yang , Jerry Zhijian JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 484 EP - 509 PY - 2022 DA - 2022/03 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2021-0085 UR - https://global-sci.org/intro/article_detail/nmtma/20361.html KW - Stokes eigenvalue problems, divergence-free, patch reconstruction, discontinuous Galerkin, mixed finite element. AB -

The discontinuous Galerkin method by divergence-free patch reconstruction is proposed for Stokes eigenvalue problems. It utilizes the mixed finite element framework. The patch reconstruction technique constructs two categories of approximation spaces. Namely, the local divergence-free space is employed to discretize the velocity space, and the pressure space is approximated by standard reconstruction space simultaneously. Benefit from the divergence-free constraint; the identical element patch serves two approximation spaces while using the element pair $\mathbb{P}^{m+1}/ \mathbb{P}^m$. The optimal error estimate is derived under the inf-sup condition framework. Numerical examples are carried out to validate the inf-sup test and the theoretical results.

Li , DiSun , ZhiyuanWang , Fengru and Yang , Jerry Zhijian. (2022). The Discontinuous Galerkin Method by Divergence-Free Patch Reconstruction for Stokes Eigenvalue Problems. Numerical Mathematics: Theory, Methods and Applications. 15 (2). 484-509. doi:10.4208/nmtma.OA-2021-0085
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