Numerical methods for spectral problems: theory and applications

P000009

Conforming and nonconforming virtual elements for eigenvalue problems

*Francesca Gardini (University of Pavia)

We analyse the conforming and nonconforming Virtual Element Method (VEM)
for the approximation of elliptic eigenvalue problems. As a model problem we consider the Laplace eigenvalue problem.
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. Moreover, we show a large set of numerical tests supporting the theoretical results, including a comparison
between the conforming and the nonconforming schemes and present some possible applications of the theory.

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Keywords

eigenvalue problems, virtual element methods, polytonal meshes