The nonconforming Virtual Element Method for eigenvalue problems

¹ Dipartimento di Matematica F. Casorati, Università di Pavia, Via Ferrata, 5 – 27100 Pavia, Italy
² Group T-5, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
³ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via R. Cozzi, 55 – 20125 Milano, Italy

^* Corresponding author: francesca.gardini@unipv.it

Received: 8 February 2018
Accepted: 29 November 2018

Abstract

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows 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 L²-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.