Fast Finite Volume Simulation of 3D Electromagnetic Problems with Highly Discontinuous Coefficients

Abstract

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies and the permeability, permittivity, and conductivity may vary significantly. 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. The present article extends methods we proposed earlier for constant permeability [E. Haber, U. Ascher, D. Aruliah, and D. Oldenburg, J. Comput. Phys., 163 (2000), pp. 150--171; D. Aruliah, U. Ascher, E. Haber, and D. Oldenburg, Math. Models Methods Appl. Sci., to appear.] to handle also problems in which the permeability is variable and may contain significant jump discontinuities. In order to address the problem of slow convergence we reformulate Maxwell's equations in terms of potentials, applying a Helmholtz decomposition to either the electric field or the magnetic field. The null space of the curl operators can then be annihilated by adding a stabilizing term, using a gauge condition, and thus obtaining a strongly elliptic differential operator. A staggered grid finite volume discretization is subsequently applied to the reformulated PDE system. This scheme works well for sources of various types, even in the presence of strong material discontinuities in both conductivity and permeability. The resulting discrete system is amenable to fast convergence of ILU-preconditioned Krylov methods. We test our method using several numerical examples and demonstrate its robust efficiency. We also compare it to the classical Yee method using similar iterative techniques for the resulting algebraic system, and we show that our method is significantly faster, especially for electric sources.