The virtual element method for linear elastodynamics models. Convergence, stability and dissipation-dispersion analysis

TL;DR

This paper develops and analyzes a virtual element method for 2D elastodynamics, demonstrating stability, convergence, and optimal error estimates, with assessments on various meshes and dispersion-dissipation properties.

Contribution

It introduces a conforming virtual element method for elastodynamics with proven stability, convergence, and optimal error estimates, including dispersion-dissipation analysis on polygonal meshes.

Findings

Optimal error estimates under h- and p-refinement

Exponential convergence observed under p-refinement

Polygonal meshes exhibit classical dispersion-dissipation behavior

Abstract

We design the conforming virtual element method for the numerical approximation of the two dimensional elastodynamics problem. We prove stability and convergence of the semi-discrete approximation and derive optimal error estimates under $h$- and $p$-refinement in both the energy and the $L^2$ norms. The performance of the proposed virtual element method is assessed on a set of different computational meshes, including non-convex cells up to order four in the $h$-refinement setting. Exponential convergence is also experimentally observed under p-refinement. Finally, we present a dispersion-dissipation analysis for both the semi-discrete and fully-discrete schemes, showing that polygonal meshes behave as classical simplicial/quadrilateral grids in terms of dispersion-dissipation properties.