Tensor Product Multigrid for Maxwell's Equation with Aligned Anisotropy

Abstract

When simulating electromagnetic phenomena in symmetric cavities, it is often possible to exploit the symmetry in order to reduce the dimension of the problem, thereby reducing the amount of work necessary for its numerical solution. Usually, this reduction leads not only to a much lower number of unknowns in the discretized system, but also changes the behaviour of the coefficients of the differential operator in an unfavourable way, usually leading to the transformed system being not elliptic with respect to norms corresponding to two-dimensional space, thus limiting the use of standard multigrid techniques. In this paper, we introduce a robust multigrid method for Maxwell's equation in two dimensions that is especially suited for coefficients resulting from coordinate transformations, i.e. that are aligned with the coordinate axes. Using a variant of the technique introduced in [5], we can prove robustness of the multigrid method for domains of tensor-product structure and coefficients depending on only one of the coordinates.