On Three-Grid Fourier Analysis for Multigrid

In this paper, we present three-grid Fourier analysis for multigrid methods. Due to the recursive structure of a multigrid iteration, this analysis can be deduced from the well-known two-grid Fourier analysis. The coarse grid correction part of multigrid algorithms can be more accurately evaluated with the three-grid analysis. We apply the analysis to several scalar equations and discretizations with an emphasis on problems with a multigrid coarse grid correction difficulty like upwind discretizations of the convection diffusion equation. The main focus lies on possible improvements by carefully chosen Galerkin operators and/or by an additional acceleration with restarted GMRES, GMRES(m). Numerical test calculations validate the theoretical predictions.