A virtual element approximation for the pseudostress formulation of the Stokes eigenvalue problem

TL;DR

This paper introduces a virtual element method for the pseudostress formulation of the Stokes eigenvalue problem, enabling accurate spectral approximation with elimination of velocity and pressure variables.

Contribution

It develops a novel VEM approach for the pseudostress formulation, ensuring correct spectral approximation and optimal error estimates.

Findings

Method accurately approximates the spectrum

The approach is spurious free

Numerical tests confirm optimal error rates

Abstract

In this paper we analyze a virtual element method (VEM) for a pseudostress formulation of the Stokes eigenvalue problem. This formulation allows to eliminate the velocity and the pressure, leading to an elliptic formulation where the only unknown is the pseudostress tensor. The velocity and pressure can be recovered by a post-process. Adapting the non-compact operator theory, we prove that our method provides a correct approximation of the spectrum and is spurious free. We prove a priori error estimates, with optimal order, which we confirm with some numerical tests.