On the spectrum of the finite element approximation of a three field formulation for linear elasticity

TL;DR

This paper analyzes the eigenvalue spectrum of a finite element three-field formulation for linear elasticity, focusing on eigenvalue distribution, approximation accuracy, and behavior near the incompressible limit.

Contribution

It provides a detailed spectral analysis of the finite element discretization for a three-field elasticity formulation, including eigenvalue distribution and parameter dependence.

Findings

Eigenvalues cluster along the positive real axis.

Spectrum depends on Lamé parameters and approaches the incompressible limit.

Discrete eigenvalues approximate continuous eigenvalues effectively.

Abstract

We continue the investigation on the spectrum of operators arising from the discretization of partial differential equations. In this paper we consider a three field formulation recently introduced for the finite element least-squares approximation of linear elasticity. We discuss in particular the distribution of the discrete eigenvalues in the complex plane and how they approximate the positive real eigenvalues of the continuous problem. The dependence of the spectrum on the Lam\'e parameters is considered as well and its behavior when approaching the incompressible limit.