We consider Maxwell's equations in the frequency domain and 3 spatial variables in parameter regimes where the quasi-static approximation applies and the permeability is constant. Usual discretizations for the electric field yield large, sparse systems of equations which are difficult to solve efficiently, especially in regions of vanishing conductivity.

The difficulty can be resolved, as for fully implicit index-2 DAEs and for certain highly oscillatory ODEs, by first applying a transformation decoupling the system appropriately. The resulting PDE is then discretized carefully using a finite volume technique, taking into account large possible jumps in conductivity. Thirdly, we solve the resulting large, sparse algebraic systems using preconditioned Krylov space methods. A particularly efficient algorithm results from the combination of BICGSTAB and a block preconditioner using an incomplete LU-decomposition of the dominant system blocks only. We demonstrate the efficacy of our method in several numerical experiments.

MINISIMPOSIUM SESSION: 13. Numerical methods for DAEs