A locking-free mixed virtual element discretization for the two dimensional elasticity eigenvalue problem

TL;DR

This paper introduces a locking-free mixed virtual element method for accurately approximating eigenvalues and eigenfunctions in 2D elasticity problems, with proven convergence and spectral correctness.

Contribution

It presents a novel virtual element discretization that is locking-free, convergent, and spectrally correct for 2D elasticity eigenvalue problems.

Findings

Method is locking-free and mesh-shape independent.

Convergence rates match theoretical predictions.

Numerical experiments confirm spectral accuracy.

Abstract

In this paper, we propose and analyze a mixed virtual element method for the approximation of the eigenvalues and eigenfunctions of the two-dimensional elasticity eigenvalue problem. Under standard assumptions on polygonal meshes, we prove the convergence of the discrete solution operator to its continuous counterpart as the mesh size tends to zero. Relying on the spectral theory of compact operators, we establish the spectral correctness of the method and derive error estimates for both eigenvalues and eigenfunctions. A series of numerical experiments is presented to support the theoretical analysis. The results confirm the predicted convergence rates and show that the method is locking-free and able to approximate the spectrum accurately, independently of the shape of the polygonal elements in the mesh.