Dept. of Computer Science Celestijnenlaan 200A B-3001 Heverlee Belgium

Abstract

We consider preconditioned Krylov subspace solvers for large, sparse, complex symmetric matrices bordered by dense blocks. Such systems arise in computational electro magnetics when e.g. the differential problem describing the magnetic field is coupled with external electrical circuit relations. In low-frequent time-harmonic Maxwell formulations in two dimensions, the governing partial differential equation is the Helmholtz equation with a complex shift. The finite element discretization of this equation leads to sparse, complex symmetric systems. The external electrical circuits constitute relations for global quantities such as the voltage drop accross or the current through the computational domain. These quantities act as a source term in the Helmholtz equation. As the external circuit relations are global relations involving integrals over (part of) the computational domain, their discretization yields dense blocks. The field-circuit coupling can be accomplished in such a way that the overall matrix remains complex symmetric.

The presence of dense blocks affects the convergence of Krylov subspace methods and hampers the immediate application of algebraic multigrid codes. This motivates the study of iterative schemes that allow to treat the discretized partial differential problem and the circuit realtions seperately.

The complex symmetry of the hybrid field-circuit coupled system allows to solve it by the Conjugate Orthogonal Conjugate Gradient (COCG) method or the Symmetric Quasi Minimal Residual (SQMR) method. A possible symmetric preconditioner is a block Jacobi scheme in which the field and circuit relations are treated as seperate blocks. In this scheme an algebraic multigrid code can be used for the discretized field equations. As the size of the circuit block is much smaller than that of the field block, the cost of solving the former is negligable compared to solving the latter. Solving the circuit block can therefore be done by a direct solver. By switching to a Gauss-Seidel scheme, the lower diagonal block in the matrix can be taken into account. The application of the Gauss-Seidel scheme as a preconditioner is more expensive than the Jacobi scheme by a multiplication of this lower diagonal block only. The cost of the block Jacobi and Gauss-Seidel schemes can be reduced by making the tolerance to which the field equations are solved dependent on the accuracy reached in the outer Krylov iteration. The resulting preconditioner chances at every Krylov iteration step. A Krylov subspace method that allows a variable preconditioner is for example Flexible GMRES (FGMRES).

Numerical experiments have shown that both the block Jacobi and the block Gauss-Seidel schemes diverge when applied as solver. The number of block Jacobi preconditioned COCG and block Gauss-Seidel preconditioned GMRES iterations is independent of the finite element mesh width. Block Gauss-Seidel preconditioned GMRES requires less iterations to converge than block Jacobi preconditioned COCG, but both are equivalent in terms of CPU time. Further research is necessary to validate the proposed schemes on models of engineering relevance.