On finite element methods for coupling eigenvalue problems

We consider second-order elliptic eigenvalue problems on a composite structure, consisting of polygonal domains in the plane, where the interaction between the domains is expressed through nonlocal coupling conditions of Dirichlet type. We study the finite element approximation without and with numerical quadrature, by adapting the operator method, outlined in [9]. In view of the error analysis, a crucial point is the definition and error estimation of a suitably modified vector Lagrange interpolant on the mesh. Compared to the results in [9], the same order of convergence in terms of the mesh parameter is achieved, however under a higher regularity assumption for the exact eigenfunctions.