A multigrid solver for two-dimensional stochastic diffusion equations

Olivier Le Maitre*, Omar Knio, B. J. Debusschere, H. N. Najm, R. G. Ghanem

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

65 Scopus citations


Steady and unsteady diffusion equations, with stochastic diffusivity coefficient and forcing term, are modeled in two dimensions by means of stochastic spectral representations. Problem data and solution variables are expanded using the Polynomial Chaos system. The approach leads to a set of coupled problems for the stochastic modes. Spatial finite-difference discretization of these coupled problems results in a large system of equations, whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time. To accelerate the convergence of the iterative technique, a multigrid method, based on spatial coarsening, is implemented. Numerical experiments show good scaling properties of the method, both with respect to the number of spatial grid points and the stochastic resolution level.

Original languageEnglish (US)
Pages (from-to)4723-4744
Number of pages22
JournalComputer Methods in Applied Mechanics and Engineering
Issue number41-42
StatePublished - Oct 10 2003


  • Diffusion equation
  • Karhunen-Loève expansion
  • Multigrid
  • Polynomial Chaos
  • Random media
  • Stochastic problem

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications


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