We investigate algorithms for computing steady state electromagnetic waves in cavities. The Maxwell equations for the strength of the electric field are solved by a mixed method with quadratic finite edge (Nédélec) elements for the field values and corresponding node-based finite elements for the Lagrange multiplier. This approach avoids so-called spurious modes which are introduced if the divergence-free condition for the electric field is not treated properly. To compute a few of the smallest positive eigenvalues and corresponding eigenmodes of the resulting large sparse matrix eigenvalue problems, two algorithms have been used: the implicitly restarted Lanczos algorithm and the Jacobi-Davidson algorithm, both with shift-and-invert spectral transformation. Two-level hierarchical basis preconditioners have been employed for the iterative solution of the resulting systems of equations.

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The sparse matrix-vector product is an important computational kernel that runs ineffectively on many computers with super-scalar RISC processors. In this paper we analyse the performance of the sparse matrix-vector product with symmetric matrices originating from the FEM and describe techniques that lead to a fast implementation. It is shown how these optimisations can be incorporated into an efficient parallel implementation using messagepassing. We conduct numerical experiments on many different machines and show that our optimisations speed up the sparse matrix-vector multiplication substantially.

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We report on a parallel implementation of the Jacobi-Davidson algorithm to compute a few eigenvalues and corresponding eigenvectors of a large real symmetric generalized matrix eigenvalue problemThe eigenvalue problem stems from the design of cavities of particle accelerators. It is obtained by the finite element discretization of the time-harmonic Maxwell equation in weak form by a combination of Nédélec (edge) and Lagrange (node) elements.We found the Jacobi-Davidson (JD) method to be a very effective solver provided that a good preconditioner is available for the correction equations that have to be solved in each step of the JD iteration. The preconditioner of our choice is a combination of a hierarchical basis preconditioner and the ML smoothed aggregation AMG preconditioner. It is close to optimal regarding iteration count.The parallel code makes extensive use of the Trilinos software framework. In our examples from accelerator physics we observe satisfactory speedups and efficiencies.

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