A mixed discontinuous Galerkin method for the time harmonic elasticity problem with reduced symmetry

TL;DR

This paper introduces a mixed discontinuous Galerkin method for solving the time-harmonic elasticity problem, allowing weak imposition of stress tensor symmetry, with proven stability and optimal error bounds validated by numerical tests.

Contribution

It presents a novel mixed discontinuous Galerkin discretization that handles weak symmetry of the stress tensor and provides rigorous stability and error analysis.

Findings

The scheme is well-posed and stable across mesh sizes and Lamé coefficients.

Optimal a-priori error bounds are established in the energy norm.

Numerical tests confirm theoretical stability and accuracy.

Abstract

The aim of this paper is to analyze a mixed discontinuous Galerkin discretization of the time-harmonic elasticity problem. The symmetry of the Cauchy stress tensor is imposed weakly, as in the traditional dual-mixed setting. We show that the discontinuous Galerkin scheme is well-posed and uniformly stable with respect to the mesh parameter $h$ and the Lam\'e coefficient $\lambda$. We also derive optimal a-priori error bounds in the energy norm. Several numerical tests are presented in order to illustrate the performance of the method and confirm the theoretical results.