Optimized Schwarz Methods for curl-curl time-harmonic Maxwell's equations

Like the Helmholtz equation, the high frequency time-harmonic Maxwell’s equations are difficult to solve by classical iterative methods. Domain decomposition methods are currently most promising: following the first provably convergent method in [4], various optimized Schwarz methods were developed over the last decade [2, 3, 10, 11, 1, 6, 13, 14, 16, 8]. There are however two basic formulations for Maxwell’s equation: the first order formulation, for which complete optimized results are known [6], and the second order, or curl-curl formulation, with partial optimization results [1, 13, 16]. We show in this paper that the convergence factors and the optimization process for the two formulations are the same. We then show by numerical experiments that the Fourier analysis predicts very well the behavior of the algorithms for a Yee scheme discretization, which corresponds to Nedelec edge elements on a tensor product mesh, in the curl-curl formulation. When using however mixed type Nedelec elements on an irregular tetrahedral mesh, numerical experiments indicate that transverse magnetic (TM) modes are less well resolved for high frequencies than transverse electric (TE) modes, and a heuristic can then be used to compensate for this in the optimization.