In this paper, we propose a novel nonstandard 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 predictive capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis could be used to estimate a priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method.

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SIAM Journal on Scientific Computing
Efficient numerical methods for deformable porous media. Application to carbon dioxide storage
Scientific Computing

Kumar, P., Rodrigo, C., Gaspar, F., & Oosterlee, K. (2019). On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients. SIAM Journal on Scientific Computing, 41(3), A1385–A1413. doi:10.1137/18M1173769