Fast Simulation of 3D Electromagnetic Problems Using Potentials

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies, the permeability is constant, the conductivity may vary significantly, and the range of frequencies is moderate. The difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems of algebraic equations that arise when discretizing Maxwell's equations in the frequency domain. We use a potential-current formulation (A, φ ,Ĵ) with a Coulomb gauge. The potentials A and φ decompose the electric field E into components in the active and null spaces of the ∇×operator. We develop a finite volume discretization on a staggered grid that naturally employs harmonic averages for the conductivity at cell faces. After discretization, we eliminate the current and the resulting large, sparse, linear system of equations has a block structure that is diagonally dominant, allowing an efficient solution with preconditioned Krylov space methods. A particularly efficient algorithm results from the combination of BICGSTAB and an incomplete LU-decomposition. We demonstrate the efficacy of our method in several numerical experiments.