Solving the Stochastic Steady-State Diffusion Problem Using Multigrid

Howard C. Elman

Department of Computer Science, University of Maryland, College Park, MD 20742

Darran Furnival

Abstract We study multigrid for the stochastic steady-state diffusion problem where the diffusion coefficient has a finite Karhunen-Lo\`eve expansion. The problem is discretized in space using linear finite elements and in the ``stochastic component'' with a polynomial chaos method. The resulting discrete system is solved using multigrid where the spatial discretization varies from grid to grid while the stochastic discretization is held constant. We establish a ``textbook'' multigrid convergence rate independent of spatial mesh size and polynomial degree of the stochastic component of the problem, and we demonstrate performance with experiments.