A Conforming virtual element approximation for the Oseen eigenvalue problem

TL;DR

This paper develops and analyzes a conforming virtual element method for accurately approximating eigenvalues and eigenfunctions of the 2D Oseen eigenvalue problem, extending virtual element techniques to fluid dynamics eigenproblems.

Contribution

It introduces a divergence-conforming virtual element approach for the Oseen eigenvalue problem and provides rigorous a priori error estimates with numerical validation.

Findings

The method achieves optimal convergence rates.

Numerical tests confirm theoretical error estimates.

The approach effectively approximates eigenvalues and eigenfunctions.

Abstract

In this paper we analyze a conforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. We consider the classic velocity-pressure formulation which allows us to consider the divergence-conforming virtual element spaces employed for the Stokes equations. Under standard assumptions on the meshes we derive a priori error estimates for the proposed method with the aid of the compact operators theory. We report some numerical tests to confirm the theoretical results.