A Virtual Element Method for the Transmission Eigenvalue Problem

TL;DR

This paper develops a virtual element method for a complex transmission eigenvalue problem, providing optimal error estimates and demonstrating effectiveness through numerical experiments.

Contribution

It introduces a $C^1$-conforming virtual element discretization for a non-selfadjoint fourth-order eigenvalue problem, with proven optimal error bounds.

Findings

Optimal order error estimates for eigenfunctions

Double order accuracy for eigenvalues

Numerical validation on various meshes

Abstract

In this paper, we analyze a virtual element method (VEM) for solving a non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a $C^1$-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-selfadjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.