A posteriori error control for discontinuous Galerkin methods for parabolic problems

TL;DR

This paper develops energy-norm a posteriori error bounds for Euler time-stepping combined with discontinuous Galerkin methods for linear parabolic problems, applicable to various schemes and illustrated with interior penalty DG.

Contribution

It introduces a general framework for a posteriori error bounds for parabolic problems using DG methods, including novel bounds for interior penalty schemes and the elliptic reconstruction technique.

Findings

Error bounds hold for a variety of DG methods

Novel a posteriori bounds derived for interior penalty DG

Framework applicable to both semi-discrete and fully discrete schemes

Abstract

We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To illustrate the method, we apply it to the interior penalty discontinuous Galerkin method, which requires the derivation of novel a posteriori error bounds. For the analysis of the time-dependent problems we use the elliptic reconstruction technique and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it.