A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations

TL;DR

This paper develops and analyzes a virtual element method for elastic vibration problems, providing error estimates and an adaptive refinement strategy using a residual-based a posteriori estimator, validated through numerical tests.

Contribution

It introduces an $H^1$-conforming VEM for elasticity spectral problems, with proven optimal error estimates and an adaptive scheme driven by a new a posteriori error estimator.

Findings

Optimal order error estimates for eigenfunctions and eigenvalues.

Reliability and efficiency of the a posteriori error estimator.

Numerical tests demonstrating the method's effectiveness.

Abstract

We present a priori and a posteriori error analysis of a Virtual Element Method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyze a variational formulation relying only on the solid displacement and propose an $H^1({\Omega})$-conforming discretization by means of VEM. Under standard assumptions on the computational domain, we show that the resulting scheme provides a correct approximation of the spectrum and prove an optimal order error estimate for the eigenfunctions and a double order for the eigenvalues. Since, the VEM has the advantage of using general polygonal meshes, which allows implementing efficiently mesh refinement strategies, we also introduce a residual-type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests that allow us to assess the performance of this approach.