Virtual Element Approximation of Eigenvalue Problems

Overview

Abstract

We will discuss the solution of eigenvalue problems associated with partial differential equations (PDE)s that can be written in the generalised form Ax = λMx, where the matrices A and/or M may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilised formulations are used for the numerical approximation of PDEs. With the help of classical numerical examples we will show that the presence of one (or both) parameters can produce unexpected results.

We will consider in particular applications arising from the use of the Virtual Element Method (VEM), where suitable parameters have to be chosen for the correct approximation. An important feature of VEM, as compared to standard FEM, is that suitable stabilising forms, depending on appropriate parameters, have to be introduced in order to guarantee stability of the approximation. Typical theoretical results state that, for given choices of the stabilisation parameters, the discrete solution converges to the continuous one with optimal order asymptotically in h or p. It is implicitly understood that the convergence behaviour depends on the choice of the parameters.

Recently, we observed [1, 2] that in the case of eigenvalue problems, the presence of the stabilising parameters may introduce artificially additional eigenmodes and we have to make sure that they will not pollute the portion of the spectrum we are interested in.
In we presented a systematic study of the eigenvalue dependence on the parameters. If the matrices A and M depend linearly on the stabilizing parameters, then in some case, the quick and easy recipe is to pick a sufficiently large stabilisation parameter for the stiffness matrix A and a small (possibly zero) stabilisation parameter for the mass matrix M.

Brief Biography

Francesca Gardini is an Associate Professor of Numerical Analysis at the Department of Mathematics, University of Pavia (Italy). She received her PhD degree in Mathematics in 2006 from the University of Pavia. Her research is concerned with the approximation of partial differential equations, with particular emphasis on the approximation of eigenvalue problems. She made significant contributions in the a priori and a posteriori error analysis of finite and mixed finite elements, virtual elements, mimetic finite differences, and adaptive methods applied to eigenvalue problems. She is the author of several papers published in highly ranked peer-reviewed International Journals.