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

We design the conforming virtual element method for the numerical approx- imation of the two-dimensional elastodynamics problem. We prove stability and convergence of the semidiscrete approximation and derive optimal error estimates under h- and p-refinement in both the energy and the L2 norms. The performance of the proposed virtual element method is assessed on a set of different computational meshes, including nonconvex 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 semidiscrete and fully discrete schemes, showing that polygonal meshes behave as classical simplicial/quadrilateral grids in terms of dispersion-dissipation properties.