Finite element approximation for 2nd order elliptic eigenvalue problems with nonlocal boundary or transition conditions

Abstract

We consider a class of eigenvalue problems (EVPs) on a bounded convex polygonal domain Ω in the plane, with nonlocal Neumann (or Robin) boundary conditions (BCs) on Γ ⊂ ∂Ω and with local Robin type BCs on ∂ΩΓ. Such problems arise in the context of some transient magnetic field computations in an electric machine. Choosing a proper space V as the space of trial- and testfunctions, we can recast the problem into the framework of abstract variational EVPs, as studied in [1]. Introducting suitable (families of) finite element subspaces V h of V, known error estimates for finite element approximations of the eigenpairs of EVPs with local BCs are found to remain valid for the present type of EVPs. We extend the analysis to EVPs in multicomponent domains with nonlocal Robin transition conditions at the internal boundaries, implying the eigenfunctions to be discontinuous at the interfaces. Such EVPs are relevant in the context of some nonperfect thermal contact problems in composite media.