Local Fourier Analysis of Multigrid for the Curl-Curl Equation

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 decou-pling the Fourier modes for edges with different orientations. Our analysis allows to quantify the smoothing properties of the considered smoothers and the convergence behavior of the considered multigrid methods. Additionally, we identify the Helmholtz splitting in Fourier space. This allows to recover several well known properties in Fourier space, such as the commutation properties of the classical Nédélec 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. 1. Introduction. The convergence of iterative methods for linear systems that arise from the discretization of a partial differential equation (PDE) can often be studied in a model problem setting by means of Fourier modes. In a local Fourier analysis (LFA) or local mode analysis, an infinite, regular grid is considered and boundary conditions are not taken into account. This type of analysis was introduced in [5]. It has been applied successfully to iterative solvers for several types of PDEs, such as the diffusion equation, the convection-diffusion equation, the Helmholtz equation, the biharmonic equation, the Stokes equations, the Oseen equations and elasticity. For comprehensive surveys, we refer the reader to [13, 15, 14]. In this paper, we develop a LFA for some multigrid methods for the curl-curl equation