Conforming virtual element approximations of the two-dimensional Stokes problem

TL;DR

This paper introduces two conforming virtual element methods for the 2D Stokes problem on polygonal meshes, demonstrating their stability, optimal convergence, and accurate divergence-free behavior.

Contribution

It presents two novel conforming VEM formulations for the Stokes problem that are stable, convergent, and effective on polygonal meshes, with verified theoretical and numerical performance.

Findings

Both formulations are inf-sup stable and convergent.

Observed convergence rates match theoretical predictions.

Zero-divergence constraint is satisfied at machine precision.

Abstract

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes.The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the $L^2$ and energy norm. We assess the effectiveness of these numerical approximations by investigating their behavior on a representative benchmark problem. The observed convergence rates are in accordance with the theoretical expectations and a weak form of the zero-divergence constraint is satisfied at the machine precision level.