Exponential convergence of the hp Virtual Element Method with corner singularities

Abstract

In the present work, we analyze the $hp$ version of Virtual Element methods for the 2D Poisson problem. We prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined, thus extending the classical choices for the decomposition in the $hp$ Finite Element framework to very general decomposition of the domain. A new stabilization for the discrete bilinear form with explicit bounds in $h$ and $p$ is introduced. Numerical experiments validate the theoretical results. We also exhibit a numerical comparison between $hp$ Virtual Elements and $hp$ Finite Elements.