An a posteriori analysis of some inconsistent, nonconforming Galerkin methods approximating elliptic problems

TL;DR

This paper provides an a posteriori analysis of inconsistent, nonconforming Galerkin methods for elliptic problems, offering new error estimates and insights into the impact of inconsistencies on scheme accuracy.

Contribution

It introduces new a posteriori error estimates for inconsistent discontinuous Galerkin schemes, extending existing results and analyzing the effects of inconsistencies on elliptic problem approximations.

Findings

Estimates match existing interior penalty DG results

New error bounds for quadrature-based DG schemes

Inconsistencies significantly affect a posteriori error analysis

Abstract

In this work we present an a posteriori analysis for classes of inconsistent, nonconforming schemes approximating elliptic problems. We show the estimates coincide with existing ones for interior penalty type discontinuous Galerkin approximations of the Laplacian and give new estimates for inconsistent discontinuous Galerkin approximation schemes of elliptic problems under quadrature approximation. We also examine the effect of inconsistencies on the a posteriori analysis of schemes applied to an unbalanced problem.