Virtual element approximation of eigenvalue problems: is the stabilization of the right hand side necessary?

TL;DR

This paper demonstrates that for certain eigenvalue problems, stabilization of the mass matrix in virtual element methods is unnecessary, simplifying the approximation process without sacrificing accuracy.

Contribution

It proves that stabilization of the mass matrix is not needed for elliptic self-adjoint eigenvalue problems with lower order VEM spaces, extending to higher order schemes.

Findings

Stabilization of the mass matrix can be omitted in certain VEM eigenvalue problems.

Numerical tests confirm the theoretical results across various mesh sequences.

Simplifies the implementation of VEM for eigenvalue problems.

Abstract

The VEM approximation of eigenvalue problems usually involves the appropriate tuning of stabilization parameters, unless self-stabilizing or stabilization-free VEM are used. In this paper we prove that for elliptic self-adjoint eigenvalue problems the stabilization of the mass matrix is not necessary when lower order standard VEM spaces are adopted. Numerical evidence shows that also for higher order schemes the same result is true on various mesh sequences.