A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

TL;DR

This paper extends the Scott-Vogelius finite element method to polygonal meshes using the Virtual Element Method for the 2-D Stokes problem, demonstrating convergence and stability properties through numerical experiments.

Contribution

It introduces a conforming virtual element formulation that generalizes Scott-Vogelius FEM for polygonal meshes in the Stokes problem context.

Findings

Converges with optimal rates on most polygonal meshes.

Fails to converge optimally on triangular and square meshes in certain cases.

Method is effective for general polygonal meshes in fluid dynamics simulations.

Abstract

The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $P_1-P_0$ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.