Numerical solution to Maxwell's equations in singular waveguides

This paper is devoted to the numerical solution of the instationary Maxwell equations in singular waveguides. The geometry is called singular, in the sense that its boundary can include reentrant corners or edges. The difficulties arise from the fact that those geometrical singularities generate, in their neighborhood, strong electromagnetic fields. Using some new theoretical and practical results on singular electromagnetic fields, we have built a method which allows to compute the time-dependent electromagnetic field. It is based on a splitting of the spaces of solutions into a two-term direct (orthogonal) sum. First, the subspace of regular fields, which coincides with the whole space of solutions, in the case of convex or smooth boundary. Second, a singular subspace, defined and characterized via the singularities of the Laplace operator. Numerical results are presented to illustrate how the frequency of the ingoing electromagnetic waves influences the singular solution in a L-shaped waveguide.