Maxwell and Lamé eigenvalues on polyhedra

Abstract

In a convex polyhedron, a part of the Lame eigenvalues with hard simple support boundary conditions does not depend on the Lame coefficients and coincides with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lame coefficients 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 finite element methods using H1 elements. Similar properties hold in polygons. We give numerical results for two L-shaped domains. Copyright © 1999 John Wiley & Sons, Ltd.