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

^** Email: belhach{at}poncelet.univ-metz.fr^*** Email: bucur{at}math.univ-metz.fr^**** Email: jmse{at}math.univ-metz.fr We study the Neumann–Laplacian eigenvalue problem in domainswith multiple cracks. We derive a mixed variational formulationwhich holds on the whole geometric domain (including the cracks)and implements efficient finite-element discretizations forthe computation of eigenvalues. Optimal error estimates aregiven and several numerical examples are presented, confirmingthe efficiency of the method. As applications, we numericallyinvestigate the behaviour of the low eigenvalues in domainswith a large number of cracks.