Basic principles of hp Virtual Elements on quasiuniform meshes

TL;DR

This paper introduces the foundational theory of hp Virtual Elements on quasiuniform meshes, providing convergence estimates and validating them through numerical experiments, with a focus on uniform polynomial degrees.

Contribution

It is the first to establish theoretical convergence estimates for hp Virtual Elements with explicit dependence on mesh size and polynomial degree.

Findings

Convergence estimates are derived for finite Sobolev regularity.

Exponential convergence is demonstrated for analytic solutions.

Numerical experiments validate the theoretical results.

Abstract

In the present paper we initiate the study of $hp$ Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size $h$ and in the polynomial degree $p$ in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included.