Fast finite volume simulation of 3D electromagnetic problems with highly discontinuous coefficients

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 en countered include handling solution discontinuities across interfaces and a ccelerating convergence of traditional iterative methods for the solution o f the linear systems of algebraic equations that arise when discretizing Ma xwell's equations in the frequency domain. The present article extends methods we proposed earlier for constant permea bility [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. Oldenbu rg, Math. Models Methods Appl. Sci., to appear.] to handle also problems in which the permeability is variable and may contain significant jump discon tinuities. In order to address the problem of slow convergence we reformula te Maxwell's equations in terms of potentials, applying a Helmholtz decompo sition to either the electric field or the magnetic field. The null space o f the curl operators can then be annihilated by adding a stabilizing term, using a gauge condition, and thus obtaining a strongly elliptic differentia l operator. A staggered grid finite volume discretization is subsequently a pplied to the reformulated PDE system. This scheme works well for sources o f various types, even in the presence of strong material discontinuities in both conductivity and permeability. The resulting discrete system is amena ble to fast convergence of ILU-preconditioned Krylov methods. We test our method using several numerical examples and demonstrate its rob ust efficiency. We also compare it to the classical Yee method using simila r iterative techniques for the resulting algebraic system, and we show that our method is significantly faster, especially for electric sources.