p- and hp- virtual elements for the Stokes problem

TL;DR

This paper analyzes the p- and hp-versions of the virtual element method for the Stokes problem on polygonal domains, establishing exponential convergence and providing explicit stabilization estimates.

Contribution

It introduces a novel interpretation of VEM for Stokes using Poisson-like VE spaces, enabling explicit analysis and exponential convergence results.

Findings

Proves exponential convergence of hp-VEM for Stokes with regular data.

Provides explicit stabilization estimates in terms of polynomial degree.

Numerical tests confirm theoretical convergence rates.

Abstract

We analyse the p- and hp-versions of the virtual element method (VEM) for the the Stokes problem on a polygonal domain. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from [7] an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.