Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks

Abstract

We study the Neumann-Laplacian eigenvalue problem in domains with multiple cracks. We derive a mixed variational formulation which holds on the whole geometric domain (including the cracks) and implements efficient finite-element discretizations for the computation of eigenvalues. Optimal error estimates are given and several numerical examples are presented, confirming the efficiency of the method. As applications, we numerically investigate the behaviour of the low eigenvalues in domains with a large number of cracks.