The nonconforming virtual element method for eigenvalue problems

TL;DR

This paper analyzes the nonconforming Virtual Element Method for elliptic eigenvalue problems, demonstrating optimal convergence rates for eigenfunctions and eigenvalues through theoretical analysis and numerical tests.

Contribution

It introduces and studies nonconforming VEM formulations for eigenvalue problems, providing convergence analysis and numerical validation.

Findings

Optimal-order error estimates for eigenfunctions

Double order convergence for eigenvalues

Numerical results confirm theoretical predictions

Abstract

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L^2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.