Avoiding stabilization terms in virtual elements for eigenvalue problems: The Reduced Basis Virtual Element Method

TL;DR

The paper introduces the Reduced Basis Virtual Element Method (rbVEM) for Laplace eigenvalue problems, eliminating stabilization terms and ensuring optimal spectral approximation through a novel combination of virtual element and reduced basis techniques.

Contribution

It presents a new stabilized-free virtual element method using reduced basis techniques for eigenvalue problems, with proven spectral accuracy.

Findings

rbVEM provides correct spectral approximation

Optimal error estimates are established

Numerical experiments confirm theoretical results

Abstract

We present the novel Reduced Basis Virtual Element Method (rbVEM) for solving the Laplace eigenvalue problem. This approach is based on the virtual element method and exploits the reduced basis technique to obtain an explicit representation of the virtual (non-polynomial) contribution to the discrete space. rbVEM yields a fully conforming discretization of the considered problem, so that stabilization terms are avoided. We prove that rbVEM provides the correct spectral approximation with optimal error estimates. Theoretical results are supplemented by an exhaustive numerical investigation.