On Computing Some of the Principal Eigenfrequencies of Cavity Resonators

Peter Arbenz

Swiss Federal Institute of Technology (ETH)
Institute of Scientific Computing
CH-8092 Zurich, Switzerland


The computation of steady state electromagnetic waves in resonant cavities is governed by the Maxwell equations. Their finite element (FEM) discretization with Nedéléc's edge elements leads to constrained symmetric matrix eigenvalue problems of the form

A x = lambda M x,   C' x = 0

Here, M is positive definite, and A, the discretization of the double-curl, is positive semidefinite. The constraint corresponds to the divergence-free condition in the continuous problem. We present and compare various approaches to compute a few of the smallest eigenvalues of this problem. The approaches differ in how the constraints are dealt with. All the approaches employ the shift-and-invert spectral transformation which implies that large sparse indefinite systems of equations have to be solved in the course of the solution of the eigenvalue problem. We use the preconditioned conjugate gradient method to solve the systems of equations. Two level preconditioners evolve naturally from the hierarchical formulation of the FEM. The theoretical behavior of the algorithms is verified by the numerical experiments.