Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamics

TL;DR

This paper develops an a posteriori analysis framework for multiphase elastodynamics models using reduced relative entropy, enabling energy-based error estimates for discontinuous Galerkin schemes involving strain and strain gradient dependencies.

Contribution

It extends the reduced relative entropy stability framework to discontinuous Galerkin methods for complex multiphase elastodynamics models.

Findings

Provides a new a posteriori error analysis approach.

Adapts the reduced relative entropy framework to DG methods.

Demonstrates stability and error control for multiphase elastodynamics.

Abstract

We give an a posteriori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics, which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in [Giesselmann 2014]. This framework allows energy type arguments to be applied to continuous functions. Since we advocate the use of discontinuous Galerkin methods we make use of two families of reconstructions, one set of discrete reconstructions [Makridakis and Nochetto 2006] and a set of elliptic reconstructions [Makridakis and Nochetto 2003] to apply the reduced relative entropy framework in this setting.