Maxwell and Lamé Eigenvalues on Polyhedra

Martin Costabel, Monique Dauge

In a convex polyhedron, a part of the Lamé eigenvalues with hard simple support boundary conditions does not depend on the Lamé coefficients and coincide with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter $s$ linked to the Lamé 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-$H^2$ 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 $H^1$ elements. Similar properties hold in polygons. We give numerical results for two L-shaped domains.

Preprint IRMAR 98-08 (Mar. 98)

Published in Mathematical Methods in the Applied Sciences 22 (1999) 243-258.

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