Spectral analysis of a mixed method for linear elasticity

TL;DR

This paper analyzes a mixed finite element method for linear elasticity eigenvalue problems, demonstrating improved approximation accuracy, stability, and efficiency through hybridization and postprocessing techniques.

Contribution

It introduces a hybridized mixed method with enhanced eigenfunction approximation and locking-free stress approximation, along with theoretical error estimates and numerical validation.

Findings

Achieves $O(h^{k+2})$ approximation for displacement eigenspace

Provides locking-free stress approximation with respect to Poisson ratio

Develops a hybridization approach for better eigenvalue approximation

Abstract

The purpose of this paper is to analyze a mixed method for linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise $(k+1)$, $k$ and $(k+1)$-th degree polynomial functions ($k\geq 1$), respectively. The numerical eigenfunction of stress is symmetric. By the discrete $H^1$-stability of numerical displacement, we prove an $O(h^{k+2})$ approximation to the $L^{2}$-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem, with proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an $O(h^2)$ initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete $H^1$-stability of numerical displacement, while only an $O(h)$ approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments.