# 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.

## Key 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.