Application of the shifted-Laplace preconditioner for iterative solution of a higher order finite element discretisation of the vector wave equation: First experiences

The analysis of electromagnetic scattering by electrically large one-side open ended cavities remains a challenge. Finite element discretisation of the vector wave equation to solve for the electric field inside the cavity leads to ill-conditioned indefinite linear systems of large dimension, a result of the requirement of a fine nearly uniform discrete sampling in the computational domain. Direct methods based on frontal solution techniques to solve the resulting linear system have been used because efficient iterative methods were not available. With the arrival of the shifted-Laplace preconditioner of Erlangga, iterative solution of the indefinite system becomes tractable. This paper discusses the modifications required for application of the shifted-Laplace preconditioner to cavity scattering and some preliminary results of this approach. The shifted-Laplace preconditioner is shown to be very effective for improving the convergence rate of the iterative solution algorithm. However, to be able to handle problems with larger number of degrees of freedom, it is necessary to include a multigrid algorithm to solve the preconditioner system, as this will allow the use of short recurrence Krylov subspace methods, as opposed to the currently employed long-recurrence method, for which the storage requirements of the Krylov basis become unpractically large. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.