A finite element method for differential eigenvalue problems with mixed classical and (semi-)periodic boundary conditions

Abstract

This paper deals with a class of elliptic eigenvalue problems (EVPs) for second-order differential equations (DEs) on a rectangular domain Ω⊂ R 2 , with periodic or semi-periodic boundary conditions (BCs) on two opposite sides of Ω. On the remaining sides, classical Robin or Dirichlet type BCs are imposed. This type of EVP is given a suitable variational formulation, which allows us to recast the EVP in the framework of abstract EVPs for symmetric, bounded, coercive bilinear forms in Hilbert spaces, leading to the existence of exact eigenpairs. From the other side the variational EVP serves as the starting point for finite element methods without and with numerical quadrature. Both triangular and rectangular meshes are considered. Moreover, we allow for the case of multiple exact eigenvalues. Well-known error estimates, established for finite element approximations of elliptic EVPs with classical BCs, are shown to remain valid for the present type of EVP too. Some attention is also paid to the computational aspects of the resulting algebraic EVP. Finally, the analysis is illustrated by a numerical example.