A pressure-robust divergence free finite element basis for the Stokes equations

Jay Chu; Xiaozhe Hu; Lin Mu; Jay Chu; Xiaozhe Hu; Lin Mu

doi:10.3934/era.2024261

Electronic Research Archive

2024, Volume 32, Issue 10: 5633-5648. doi: 10.3934/era.2024261

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A pressure-robust divergence free finite element basis for the Stokes equations

Jay Chu ¹,
Xiaozhe Hu ²,
Lin Mu ^{3
,
,}

1.
Departmant of Mathematics, National Tsing Hua University, Taiwan
2.
Department of Mathematics, Tufts University, Medford, MA 02155, USA
3.
Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Received: 24 July 2024 Revised: 16 September 2024 Accepted: 07 October 2024 Published: 16 October 2024

This paper considered divergence-free basis methods to solve the viscous Stokes equations. A discrete divergence-free subspace was constructed to reduce the saddle point problem of the Stokes problem to a smaller-sized symmetric and positive definite system solely depending on the velocity components. Then, the system could decouple the unknowns in velocity and pressure and solve them independently. However, such a scheme may not ensure an accurate numerical solution to the velocity. In order to obtain satisfactory accuracy, we used a velocity reconstruction technique to enhance the divergence-free scheme to achieve the desired pressure and viscosity robustness. Numerical results were presented to demonstrate the robustness and accuracy of this discrete divergence-free method.
- finite element methods,
- the viscous Stokes equations,
- divergence-free basis,
- pressure robustness
Citation: Jay Chu, Xiaozhe Hu, Lin Mu. A pressure-robust divergence free finite element basis for the Stokes equations[J]. Electronic Research Archive, 2024, 32(10): 5633-5648. doi: 10.3934/era.2024261

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Abstract

This paper considered divergence-free basis methods to solve the viscous Stokes equations. A discrete divergence-free subspace was constructed to reduce the saddle point problem of the Stokes problem to a smaller-sized symmetric and positive definite system solely depending on the velocity components. Then, the system could decouple the unknowns in velocity and pressure and solve them independently. However, such a scheme may not ensure an accurate numerical solution to the velocity. In order to obtain satisfactory accuracy, we used a velocity reconstruction technique to enhance the divergence-free scheme to achieve the desired pressure and viscosity robustness. Numerical results were presented to demonstrate the robustness and accuracy of this discrete divergence-free method.

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