A method for the forward modelling of 3D electromagnetic quasi-static problems

We present a solution method for solving electromagnetic problems in three dimensions in parameter regimes where the quasi-static approximation applies and the permeability is constant. Firstly, by using a potential formulation with a Coulomb gauge, we circumvent the ill-posed problem in regions of vanishing conductivity, obtaining a system of elliptic, weakly coupled differential equations. The system thus derived is strongly elliptic, which leads to reliable discretizations. Secondly, we derive a robust finite-volume discretization. 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. Keywords: vector potential, Coulomb gauge, Maxwell's equations, finite volume, Krylov methods, pre..