On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
For researchers solving PDEs with heterogeneous coefficients, this work provides a predictive tool for multigrid performance, though it is an incremental extension of existing LFA techniques.
The paper introduces a non-standard Local Fourier Analysis (LFA) variant that accurately predicts multigrid convergence for problems with random and jumping coefficients, demonstrated on a cell-centered multigrid method for flow in random porous media. The analysis enables a-priori estimation of solution time for uncertainty quantification problems using multigrid multilevel Monte Carlo.
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the prediction capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis helps us to estimate a-priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method.