A virtual element method for the Steklov eigenvalue problem allowing small edges

TL;DR

This paper analyzes the impact of small edges in polygonal meshes on the virtual element method for the Steklov eigenvalue problem, demonstrating accurate spectrum approximation and optimal error estimates despite mesh irregularities.

Contribution

It introduces a virtual element method that tolerates arbitrarily small edges in meshes and proves its spectral approximation accuracy under weaker mesh assumptions.

Findings

The method accurately approximates the spectrum despite small edges.

Optimal error estimates are established for eigenfunctions.

Numerical tests confirm theoretical predictions.

Abstract

The aim of this paper is to analyze the influence of small edges in the computation of the spectrum of the Steklov eigenvalue problem by a lowest order virtual element method. Under weaker assumptions on the polygonal meshes, which can permit arbitrarily small edges with respect to the element diameter, we show that the scheme provides a correct approximation of the spectrum and prove optimal error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report some numerical tests supporting the theoretical results.