On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients

TL;DR

This paper introduces a new local Fourier analysis method tailored for multigrid methods applied to PDEs with complex coefficients, improving convergence predictions for uncertain and heterogeneous media.

Contribution

A novel LFA variant based on a specific Fourier basis is proposed, enhancing accuracy in predicting multigrid convergence for problems with random and jumping coefficients.

Findings

Accurately predicts multigrid convergence in challenging PDE problems.

Demonstrates utility through benchmark tests with random porous media.

Estimates solution times for uncertainty quantification tasks.

Abstract

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.