A DG Method for the Stokes Equations on Tensor Product Meshes with

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}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[P_k]^d-P_{k-1}$$\end{document}

element on the tensor product meshes. Comparing to the previous stability proof with

{[Q_k]}^d-Q_{k-1}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[Q_k]^d-Q_{k-1}$$\end{document}

discontinuous finite elements in the existing references, our first contribution is to extend the formal proof to the

{[P_k]}^d-P_{k-1}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[P_k]^d-P_{k-1}$$\end{document}

discontinuous elements on the tensor product meshes. Numerical inf-sup tests have been performed to compare

Q_k

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and

P_k

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_k$$\end{document}

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}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[P_k]^d-P_{k-1}$$\end{document}

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.

Keywords

Finite element / Discontinuous Galerkin (DG) method / Tensor product mesh / Enhancement of pressure-robustness

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Funding

Simons Foundation(964995)