A posteriori error estimates for discontinuous Galerkin method to the elasticity problem

TL;DR

This paper develops a posteriori error estimates for the discontinuous Galerkin method applied to time-dependent linear elasticity problems, using stationary elasticity reconstruction to connect stationary and dynamic error bounds.

Contribution

It introduces a novel approach to derive a posteriori error bounds for DG methods in elasticity, employing stationary problem reconstruction and duality/energy techniques.

Findings

Derived error bounds for semi-discrete and fully discrete schemes

Implemented backward-Euler scheme with space-time reconstruction

Applicable to various DG methods for elasticity problems

Abstract

This work concerns with the discontinuous Galerkin (DG)method for the time-dependent linear elasticity problem. We derive the a posteriori error bounds for semi-discrete and fully discrete problems, by making use of the stationary elasticity reconstruction technique which allows to estimate the error for time-dependent problem through the error estimation of the associated stationary elasticity problem. To this end, to derive the error bound for the stationary problem, we present two methods to obtain two different a posteriori error bounds, by $L^2$ duality technique and via energy norm. For fully discrete scheme, we make use of the backward-Euler scheme and an appropriate space-time reconstruction. The technique here can be applicable for a variety of DG methods as well.