On the validity of the local Fourier analysis

TL;DR

This paper extends the applicability of local Fourier analysis (LFA) to a broader class of problems, demonstrating that it can accurately predict convergence factors beyond rectangular domains with periodic boundary conditions.

Contribution

The authors prove that LFA provides exact convergence factors for a wider class of problems using the Fourier method, beyond traditional rectangular periodic domains.

Findings

LFA accurately predicts convergence factors for a broader problem class

Extension of LFA validity beyond rectangular periodic domains

Mathematical proof supporting the extended applicability

Abstract

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. 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.