Multigrid method for Maxwell's equations

In this paper we are concerned with the efficient solution of discrete vari ational problems related to the bilinear form (curl ., curl .) (L2(Omega)) + (.,.) (L2(Omega)) defined on H-o (curl; Omega). This is a core task in th e time-domain simulation of electromagnetic fields, if implicit timesteppin g is employed. We rely on Nedelec's H (curl; Omega)-conforming finite eleme nts (edge elements) to discretize the problem. We construct a multigrid method for the fast iterative solution of the resu lting linear system of equations. Since proper ellipticity of the bilinear form is confined to the complement of the kernel of the curl operator, Helm holtz decompositions are the key to the design of the algorithm: N (curl) a nd its complement N(curl)(perpendicular to) require separate treatment. Bot h can be tackled by nodal multilevel decompositions, where for the former t he splitting is set in the space of discrete scalar potentials. Under certain assumptions on the computational domain and the material func tions, a rigorous proof of the asymptotic optimality of the multigrid metho d can be given, which shows that convergence does not deteriorate on very f ine grids. The results of numerical experiments confirm the practical effic iency of the method.