An a posteriori error estimator for discontinuous Galerkin methods for non-stationary convection-diffusion problems

TL;DR

This paper develops a robust a posteriori error estimator for discontinuous Galerkin methods applied to non-stationary convection-diffusion problems, enabling effective adaptive algorithms with proven optimal convergence rates.

Contribution

It introduces a Péclet number robust a posteriori error estimator for DG discretizations of non-stationary convection-diffusion problems and implements an adaptive algorithm demonstrating optimal convergence.

Findings

Estimator is robust with respect to Péclet number

Adaptive algorithm achieves optimal convergence rates

Numerical experiments confirm theoretical robustness

Abstract

This work is concerned with the derivation of a robust a posteriori error estimator for a discontinuous Galerkin method discretisation of linear non-stationary convection-diffusion initial/boundary value problems and with the implementation of a corresponding adaptive algorithm. More specifically, we derive a posteriori bounds for the error in the $L^2(H^1)$-type norm for an interior penalty discontinuous Galerkin (dG) discretisation in space and a backward Euler discretisation in time. An important feature of the estimator is robustness with respect to the P\'{e}clet number of the problem which is verified in practice by a series of numerical experiments. Finally, an adaptive algorithm is proposed utilising the error estimator. Optimal rate of convergence of the adaptive algorithm is observed in a number of test problems.