Fourier mode analysis of multigrid methods for partial differential equations with random coefficients

Abstract

Partial differential equations with random coefficients appear for example in reliability problems and uncertainty propagation models. Various approaches exist for computing the stochastic characteristics of the solution of such a differential equation. In this paper, we consider the spectral expansion approach. This method transforms the continuous model into a large discrete algebraic system. We study the convergence properties of iterative methods for solving this discretized system. We consider one-level and multi-level methods. The classical Fourier mode analysis technique is extended towards the stochastic case. This is done by taking the eigenstructure into account of a certain matrix that depends on the random structure of the problem. We show how the convergence properties depend on the particulars of the algorithm, on the discretization parameters and on the stochastic characteristics of the model. Numerical results are added to illustrate some of our theoretical findings.