Maxwell and Lame eigenvalues on polyhedra

In a convex polyhedron, a part of the Lame eigenvalues with hard simple sup port boundary conditions does not depend on the Lame: coefficients and coin cides with the Maxwell eigenvalues. The other eigenvalues depend linearly o n a parameter s linked to the Lame coefficients and the associated eigenmod es are the gradients of the Laplace-Dirichlet eigenfunctions. In a non-conv ex polyhedron, such a splitting of the spectrum disappears partly or comple tely, in relation with the non-H-2 singularities of the Laplace-Dirichlet e igenfunctions. From the Maxwell equations point of view, this means that in a non-convex polyhedron, the spectrum cannot be approximated by finite ele ment methods using H-1 elements. Similar properties hold in polygons. We gi ve numerical results for two L-shaped domains. Copyright (C) 1999 John Wile y & Sons, Ltd.