A posteriori error bounds for discontinuous Galerkin methods for quasilinear parabolic problems

TL;DR

This paper develops a posteriori error bounds in the energy norm for the $hp$-version interior penalty discontinuous Galerkin method applied to quasilinear parabolic problems, using elliptic reconstruction techniques.

Contribution

It introduces a novel a posteriori error estimation framework for IPDG methods on quasilinear parabolic problems, focusing on the semidiscrete case.

Findings

Derived energy-norm error bounds for IPDG discretizations.

Utilized elliptic reconstruction framework for error analysis.

Focused on strictly monotone quasilinear parabolic problems.

Abstract

We derive a posteriori error bounds for a quasilinear parabolic problem, which is approximated by the $hp$-version interior penalty discontinuous Galerkin method (IPDG). The error is measured in the energy norm. The theory is developed for the semidiscrete case for simplicity, allowing to focus on the challenges of a posteriori error control of IPDG space-discretizations of strictly monotone quasilinear parabolic problems. The a posteriori bounds are derived using the elliptic reconstruction framework, utilizing available a posteriori error bounds for the corresponding steady-state elliptic problem.