On a variational approximation method for 2nd order eigenvalue problems in a multi–component domain with nonlocal dlrichlet transition conditions

Abstract

We consider a class of eigenvalue problems (EVPs) on a bounded multi-component domain Ω in the plane, consisting of a number of convex polygonal subdomains. Classical mixed Robin-Dirichlet conditions are imposed on the outer boundaries, while on the interfaces between two subdomains we impose nonlocal transition conditions of Dirichlet type. This problem extends the single-component problem, treated in [4]. The aim of the paper is twofold. First, by passing to a product Sobolev space setting, we recast the problem into the framework of abstract variational EVPs, studied e.g.in [7]. This allows us to infer the existence of exact eigenpairs, showing some useful properties. Secondly, the variational EVP may serve as the starting point for setting up variational (internal) approximation methods, such as (conforming) finite element methods (FEMs). For this purpose the proof of a density theorem is crucial. From here on, for clarity in the exposition, we restrict ourselves to a domain Ω composed of 4 rectangles. Introducing suitable (families of) finite element spaces V h1, similar convergence properties and error estimates as in [4] are obtained for the finite element approximations of the eigenpairs, both without and with numerical quadrature. The error analysis will heavily rest upon the properties of a deliberately introduced imperfect Lagrange interpolant on each of the subdomains. Finally, we briefly deal with the corresponding algebraic EVP, identifying a proper basis of the product space V h.