We consider three-dimensional electromagnetic problems that arise in forward-modelling of Maxwell's equations in the frequency domain. Traditional formulations and discretizations of Maxwell's equations for this class of problems leads to a large, sparse, system of linear algebraic equations that is difficult to solve. That is, iterative methods applied to the linear system are slow to converge which is a major obstacle in solving practical inverse problems.

A finite volume discretization on a staggered grid derived from a potential-current formulation of Maxwell's equations with a block structure that is diagonally dominant. The block structure of the linear system stems directly from the choice of potential formulation for Maxwell's frequency domain equations. The system obtained is strongly elliptic which in turn suggests various strategies for preconditioning Krylov space iterations used to solve the linear system.

A constant-coefficient, periodic, local-mode analysis common suggests that exact inversion of the diagonal blocks can be used as an effective preconditioning strategy. In particular, local-mode analysis shows that the preconditioned matrix has eigenvalues range and of the $l_2$-conditioned number. For Hermitian problems, this implies Krylov-subspace iterations will achieve rates of convergence independent of the mesh size; that is, a fixed reduction of the relative residual is attainable in a constant number of iterations regardless of the degree to which the mesh is refined. The analysis extends readily to the case where a single multigrid V-cycle is applied for preconditioning rather than invertinf the diagonal blocks exactly giving similar convergence properties. The multigrid preconditioning strategy is tested within Krylov-subspace iterations in a variety of experiments. In spite of the non-Hermitian character of the underlying linear system, the numerical experiments support theoretical predictions and demonstrate the efficacy of multigrid preconditioning for low-to-moderate frequency electromagnetic simulations