Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Abstract
We deal with a class of elliptic eigenvalue problems (EVPs)
on a rectangle Ω ⊂ R^2 , with periodic or semi–periodic boundary conditions
(BCs) on ∂Ω. First, for both types of EVPs, we pass to a proper variational
formulation which is shown to fit into the general framework of abstract
EVPs for symmetric, bounded, strongly coercive bilinear forms in Hilbert
spaces, see, e.g., [13, §6.2]. Next, we consider finite element methods (FEMs)
without and with numerical quadrature. The aim of the paper is to show
that well–known error estimates, established for the finite element approximation
of elliptic EVPs with classical BCs, hold for the present types of
EVPs too. Some attention is also paid to the computational aspects of the
resulting algebraic EVP. Finally, the analysis is illustrated by two non-trivial
numerical examples, the exact eigenpairs of which can be determined
