Local Fourier Analysis of Multigrid for the Curl-Curl Equation

Abstract

We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwell's equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk, and Winther are considered. The key to our approach is the identification of two-dimensional eigenspaces of the discrete curl-curl problem by decoupling the Fourier modes for edges with different orientations. This procedure is used to quantify the smoothing properties of the considered smoothers and the convergence behavior of the multigrid methods. Additionally, we identify the Helmholtz splitting in Fourier space. This allows several well known properties to be recovered in Fourier space, such as the commutation properties of the classical Nedelec prolongator and the equivalence of the curl-curl operator and the vector Laplacian for divergence-free vectors. We show how the approach used in this paper can be generalized to two- and three-dimensional problems in $H$(curl) and $H$(div) and to other types of regular meshes.