A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems

TL;DR

This paper develops a posteriori error bounds for fully discrete hp-discontinuous Galerkin time-stepping methods applied to linear parabolic PDEs, using a novel space-time reconstruction approach that is flexible and adaptable to various estimators.

Contribution

It introduces a new space-time reconstruction technique for deriving computable a posteriori error bounds in fully discrete hp-DG methods for parabolic problems.

Findings

Derivation of error bounds in multiple norms including $L_{ abla}(I;L_2( abla))$ and $L_2(I;H^1( abla))$.

Flexibility in the choice of elliptic error estimators.

Clear framework for mesh-change estimators.

Abstract

We consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an $hp$-version discontinuous Galerkin (DG) time stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds resulting, for instance, in concrete bounds in $L_{\infty}(I;L_2(\Omega))$- and $L_{2}(I;H^{1}(\Omega))$-type norms when $I$ is the temporal and $\Omega$ the spatial domain for the PDE. We base our methodology for the analysis on a novel space-time reconstruction approach. Our approach is flexible as it works for any type of elliptic error estimator and leaves their choice of up to the user. It also allows exhibits mesh-change estimators in a clear an concise way. We also show how our approach allows the derivation of such bounds in the $H^1(I;H^{-1}(\Omega))$ norm.