A virtual element method for the Steklov eigenvalue problem

Abstract

The aim of this paper is to develop a virtual element method for the two-dimensional Steklov eigenvalue problem. We propose a discretization by means of the virtual elements presented in [L. Beirão da Veiga et al., Basic principles of virtual element methods, Math. Models Methods Appl. Sci.23 (2013) 199–214]. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates for the eigenfunctions and a double order for the eigenvalues. We also prove higher-order error estimates for the computation of the eigensolutions on the boundary, which in some Steklov problems (computing sloshing modes, for instance) provides the quantity of main interest (the free surface of the liquid). Finally, we report some numerical tests supporting the theoretical results.

Divergence free virtual elements for the stokes problem on polygonal meshes

“…Moreover, it clearly holds Π (29) and the subsequent discussion), it follows that the operator Π ∇,K k is computable in terms of the degrees of freedom D V .…”

“…The proof follows the guidelines of Proposition 4.2 in [29]. For each polygon K ∈ T h , let us consider the triangulation T K h of K obtained by joining each vertex of K with the center of the ball with respect to which K is star-shaped.…”

BDDC and FETI-DP for the virtual element method

“…As far as Proposition 3.5, its proof relies on the existence of a Scott-Zhang type operator satisfying an estimate of the form (3.9). The construction of such an operator, as proposed in [52] for the case here considered, carries over to other H 1 conforming VEM spaces, yielding (3.9) with constants possibly depending on k. It is in fact not difficult to verify that adapting such a construction to different definitions of the local space V K,k yields an operator that locally preserves polynomials of order k (provided of course that they are contained in the local space). Remark 4.4.…”

Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

“…Indeed, by avoiding the explicit construction of the local basis functions, the VEM can easily handle general polygons/polyhedrons without complex integrations on the element (see [9] for details on the coding aspects of the method). The Virtual Element Method has been applied successfully in a large range of problems, see for instance [1,2,7,8,9,12,15,16,17,20,23,25,28,35,39,40,41,47,48].…”