Maxwell and Lam E Eigenvalues on Polyhedra

e Eigenvalues on Polyhedra Martin COSTABEL & Monique DAUGE Abstract. In a convex polyhedron, a part of the Lam e eigenvalues with hard simple support boundary conditions does not depend on the Lam e coe cients and coincide with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lam e coe cients and the associated eigenmodes are the gradients of the Laplace-Dirichlet eigenfunctions. In a non-convex polyhedron, such a splitting of the spectrum disappears partly or completely, in relation with the non-H2 singularities of the Laplace-Dirichlet eigenfunctions. From the Maxwell equations point of view, this means that in a non-convex polyhedron, the spectrum cannot be approximated by nite element methods using H1 elements. Similar properties hold in polygons. We give numerical results for two L-shaped domains.