# Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

## TL;DR

This paper analyzes the stability and exponential convergence of spectral mixed discontinuous Galerkin methods for 3D linear elasticity problems on anisotropic meshes, especially near incompressible limits.

## Contribution

It provides a computational study on the stability and convergence of mixed DG schemes for elasticity on complex meshes, including robustness near incompressibility.

## Key findings

- Methods are stable with respect to the Poisson ratio.
- Numerical evidence shows exponential convergence.
- Schemes effectively handle singularities on polyhedral domains.

## Abstract

We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lam\'e system of linear elasticity in polyhedral domains in $\mathbb{R}^3$. In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we illustrate numerically that the proposed mixed DG schemes converge exponentially in a natural DG norm.