A finite element method for elliptic eigenvalue problems in a multi-component domain in 2D with non-local Dirichlet transition conditions

Abstract

In this paper we consider a class of eigenvalue problems (EVPs) on a bounded multi-component domain in the plane, which consists of a number of convex polygonal subdomains. On the outer boundaries classical mixed Robin–Dirichlet conditions hold, while we impose nonlocal transition conditions (TCs) of Dirichlet-type on the interfaces between two subdomains. First, we state the variational formulation of this problem. This variational EVP then serves as the starting point for internal approximation methods such as finite element methods (FEMs), developed here. The error analysis involved mainly rests upon the properties of a deliberately defined imperfect Lagrange interpolant. Considerable attention is also paid to a crucial density result and to the computational aspects. This paper extends the results of De Schepper and Van Keer (Numer. Funct. Anal. Optim. 19 (9&10) (1998)), where the finite element analysis was restricted to the case of a rectangle consisting of four rectangular subdomains. Finally, the analysis is illustrated by a numerical example.