Analysis of Discontinuous Galerkin Methods using Mesh-Dependent Norms and Applications to Problems with Rough Data

TL;DR

This paper establishes the stability and error bounds of a discontinuous Galerkin method for second order elliptic problems using mesh-dependent norms, with applications to problems involving rough data where traditional solutions are inadequate.

Contribution

It proves inf-sup stability and a priori error bounds for DG schemes in mesh-dependent norms, extending analysis to problems with rough source terms.

Findings

Proves stability of DG methods in mesh-dependent norms for all dimensions.

Derives a priori error bounds in these norms.

Demonstrates quasi-optimal error control for problems with rough data.

Abstract

We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine a problem with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control.