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

TL;DR

This paper develops a reduced relative entropy framework to analyze a semi-discrete discontinuous Galerkin scheme for multiphase elastodynamics, providing optimal bounds for strain and suboptimal bounds for velocity.

Contribution

It introduces a novel a priori analysis method using reduced relative entropy for multiphase elastodynamics models with strain gradient dependence.

Findings

Proves optimal bounds for strain in the model.

Establishes suboptimal bounds for velocity.

Validates the stability of the proposed numerical scheme.

Abstract

We give an a priori 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 [Gie14]. We prove optimal bounds for the strain in an appropriate norm and suboptimal bounds for the velocity.