(untitled) (Journal of Computational Mathematics, 2019)
Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such methods. When other boundary conditions are considered, however, this analysis was judged as been heuristic, with limited capabilities in predicting multigrid convergence rates. In this work, using the Fourier method, we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems, some of which cannot be handled by the traditional rigorous Fourier analysis.