A DG Method for the Stokes Equations on Tensor Product Meshes with [Pk]d-Pk-1 Element

doi:10.1007/s42967-022-00243-9

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (4): 2431-2454.doi: 10.1007/s42967-022-00243-9

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A DG Method for the Stokes Equations on Tensor Product Meshes with [P_k]^d-P_k-1 Element

Lin Mu¹, Xiu Ye², Shangyou Zhang³, Peng Zhu⁴

1 Department of Mathematics, University of Georgia, Athens 30602, GA, USA;
2 Department of Mathematics, University of Arkansas at Little Rock, Little Rock 72204, AR, USA;
3 Department of Mathematical Sciences, University of Delaware, Newark 19716, DE, USA;
4 College of Data Science, Jiaxing University, Jiaxing 314001, Zhejiang, China

收稿日期:2022-04-21 修回日期:2022-11-01 接受日期:2022-12-06 发布日期:2024-12-20
通讯作者: Lin Mu,E-mail:linmu@ualr.edu;Xiu Ye,E-mail:xxye@ualr.edu;Shangyou Zhang,E-mail:szhang@udel.edu;Peng Zhu,E-mail:pzh@zjxu.edu.cn E-mail:linmu@ualr.edu;xxye@ualr.edu;szhang@udel.edu;pzh@zjxu.edu.cn
基金资助:
The work of the first author is partially supported by the Simons Foundation Grant.

A DG Method for the Stokes Equations on Tensor Product Meshes with [P_k]^d-P_k-1 Element

Lin Mu¹, Xiu Ye², Shangyou Zhang³, Peng Zhu⁴

1 Department of Mathematics, University of Georgia, Athens 30602, GA, USA;
2 Department of Mathematics, University of Arkansas at Little Rock, Little Rock 72204, AR, USA;
3 Department of Mathematical Sciences, University of Delaware, Newark 19716, DE, USA;
4 College of Data Science, Jiaxing University, Jiaxing 314001, Zhejiang, China

Received:2022-04-21 Revised:2022-11-01 Accepted:2022-12-06 Published:2024-12-20
Contact: Lin Mu,E-mail:linmu@ualr.edu;Xiu Ye,E-mail:xxye@ualr.edu;Shangyou Zhang,E-mail:szhang@udel.edu;Peng Zhu,E-mail:pzh@zjxu.edu.cn E-mail:linmu@ualr.edu;xxye@ualr.edu;szhang@udel.edu;pzh@zjxu.edu.cn

摘要/Abstract

摘要： We consider the mixed discontinuous Galerkin (DG) finite element approximation of the Stokes equation and provide the analysis for the [P_k]^d-P_k-1 element on the tensor product meshes. Comparing to the previous stability proof with[Q_k]^d-Q_k-1 discontinuous finite elements in the existing references, our first contribution is to extend the formal proof to the [P_k]^d-P_k-1 discontinuous elements on the tensor product meshes. Numerical infsup tests have been performed to compare Q_k and P_k types of elements and validate the well-posedness in both settings. Secondly, our contribution is to design the enhanced pressure-robust discretization by only modifying the body source assembling on [P_k]^d-P_k-1 schemes to improve the numerical simulation further. The produced numerical velocity solution via our enhancement shows viscosity and pressure independence and thus outperforms the solution produced by standard discontinuous Galerkin schemes. Robustness analysis and numerical tests have been provided to validate the scheme's robustness.

关键词: Finite element, Discontinuous Galerkin(DG)method, Tensor product mesh, Enhancement of pressure-robustness

Abstract: We consider the mixed discontinuous Galerkin (DG) finite element approximation of the Stokes equation and provide the analysis for the [P_k]^d-P_k-1 element on the tensor product meshes. Comparing to the previous stability proof with[Q_k]^d-Q_k-1 discontinuous finite elements in the existing references, our first contribution is to extend the formal proof to the [P_k]^d-P_k-1 discontinuous elements on the tensor product meshes. Numerical infsup tests have been performed to compare Q_k and P_k types of elements and validate the well-posedness in both settings. Secondly, our contribution is to design the enhanced pressure-robust discretization by only modifying the body source assembling on [P_k]^d-P_k-1 schemes to improve the numerical simulation further. The produced numerical velocity solution via our enhancement shows viscosity and pressure independence and thus outperforms the solution produced by standard discontinuous Galerkin schemes. Robustness analysis and numerical tests have been provided to validate the scheme's robustness.

Key words: Finite element, Discontinuous Galerkin(DG)method, Tensor product mesh, Enhancement of pressure-robustness

Lin Mu, Xiu Ye, Shangyou Zhang, Peng Zhu. A DG Method for the Stokes Equations on Tensor Product Meshes with [P_k]^d-P_k-1 Element[J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2431-2454.

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A DG Method for the Stokes Equations on Tensor Product Meshes with [P_k]^d-P_k-1 Element

A DG Method for the Stokes Equations on Tensor Product Meshes with [P_k]^d-P_k-1 Element

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