Mixed discontinuous Galerkin approximation of the elasticity eigenproblem

TL;DR

This paper presents a novel mixed discontinuous Galerkin method for the elasticity eigenproblem that accurately approximates the spectrum despite nonconformity and non-compactness, supported by theoretical analysis and numerical validation.

Contribution

It introduces a new discontinuous Galerkin scheme for elasticity eigenproblems with reduced symmetry, including spectral analysis and error estimates.

Findings

The scheme correctly approximates the spectrum.

Asymptotic error estimates are established.

Numerical tests confirm theoretical predictions.

Abstract

We introduce a discontinuous Galerkin method for the mixed formulation of the elasticity eigenproblem with reduced symmetry. The analysis of the resulting discrete eigenproblem does not fit in the standard spectral approximation framework since the underlying source operator is not compact and the scheme is nonconforming. We show that the proposed scheme provides a correct approximation of the spectrum and prove asymptotic error estimates for the eigenvalues and the eigenfunctions. Finally, we provide several numerical tests to illustrate the performance of the method and confirm the theoretical results.