A robust discontinuous Galerkin scheme on anisotropic meshes

TL;DR

This paper introduces a new symmetric interior penalty discontinuous Galerkin scheme that remains robust and accurate on anisotropic meshes without requiring shape-regularity conditions, supported by theoretical analysis and numerical experiments.

Contribution

A novel DG scheme with a modified penalty term that is robust on anisotropic meshes, removing the need for shape-regularity assumptions.

Findings

The new scheme inherits all properties of standard DG methods.

The scheme is robust on anisotropic meshes.

Numerical experiments confirm theoretical error estimates.

Abstract

Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.