Anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation

TL;DR

This paper introduces an anisotropic weakly over-penalised symmetric interior penalty discontinuous Galerkin method for the Stokes equation, providing new proofs, stability analysis on anisotropic meshes, and error estimates, validated by numerical experiments.

Contribution

The paper presents a novel proof for the consistency term and demonstrates inf-sup stability of the Stokes element on anisotropic meshes, along with error estimates.

Findings

Inf-sup stability on anisotropic meshes confirmed.

Error estimates in energy norm derived.

Numerical experiments validate theoretical results.

Abstract

In this study, we investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation {on convex domains}. Our approach is a simple discontinuous Galerkin method similar to the Crouzeix--Raviart finite element method. As our primary contribution, we show a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart--Thomas finite element space and a discontinuous space. While inf-sup stable schemes of the discontinuous Galerkin method on shape-regular mesh partitions have been widely discussed, our results show that the Stokes element satisfies the inf-sup condition on anisotropic meshes. Furthermore, we provide an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we compare calculation results for standard and anisotropic mesh partitions.