The p- and hp-versions of the virtual element method for elliptic eigenvalue problems

Abstract

We discuss the p-and the hp-versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schrödinger equation with a pseudo-potential term. We present in details the analysis of the p-version of the method, proving exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing hp-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the hp-spaces.

Approximation of PDE eigenvalue problems involving parameter dependent matrices

“…This is a typical situation found in applications where elliptic partial differential equations are approximated by schemes that require suitable parameters to be tuned (for consistency and/or stability reasons). In this paper we discuss in particular applications arising from the use of the Virtual Element Method (VEM), see [5,13,15,16,[20][21][22], where suitable parameters have to be chosen for the correct approximation. Similar situations are present, for instance, when a parameter-dependent stabilization is used for the approximation of discontinuous Galerkin formulations and when a penalty term is added to the discretization of the eigenvalue problem associated with Maxwell's equations [2, 6-8, 10-12, 23, 26] In general, it may be not immediate to describe how the matrices and depend on the given parameters.…”

Approximation of PDE eigenvalue problems involving parameter dependent matrices

“…This is a typical situation found in applications where elliptic partial differential equations are approximated by schemes that require suitable parameters to be tuned (for consistency and/or stability reasons). In this paper we discuss in particular applications arising from the use of the Virtual Element Method (VEM), see [9,4,7,10,11,6,12], where suitable parameters have to be chosen for the correct approximation.…”

A virtual element approximation for the pseudostress formulation of the Stokes eigenvalue problem

“…In the present paper we introduce a high order VEM in order to solve problem (1.1) with the pseuodstress formulation introduced in [27]. Several papers deal with the Stokes and Navier Stokes problems, implementing the VEM in order to approximate the velocity and pressure considering different formulations (see for instance [1,3,4,6,7,8,11,12,15,16,17,18,21,22,24,28,29,30]). In particular, in [8] the authors analyze rigorously a VEM for the steady Stokes problem, introducing the pseudostress tensor which leads to a mixed formulation where the main unknowns are the velocity field and the pseudostress.…”