Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs

Since their popularization in the late 1970s and early 1980s, multigrid methods have been a central tool in the numerical solution of the linear and nonlinear systems that arise from the discretization of many PDEs. In this paper, we present a local Fourier analysis (LFA, or local mode analysis) framework for analyzing the complementarity between relaxation and coarse-grid correction within multigrid solvers for systems of PDEs. Important features of this analysis framework include the treatment of arbitrary finite-element approximation subspaces, leading to discretizations with staggered grids, and overlapping multiplicative Schwarz smoothers. The resulting tools are demonstrated for the Stokes, curl–curl, and grad–div equations.