International Congress on Computational and Applied Mathematics edition:13 location:Ghent, Belgium date:7-11 July 2008
Solving partial differential equations (PDEs) that contain random or stochastic parameters, is in general a difficult and computationally intensive task. Typically Monte Carlo simulations are applied. However, they may require a huge amount of computational work. Therefore, alternative solution approaches have been proposed. Amongst these, the stochastic Galerkin finite element method has received
a lot of attention in recent years. This method transforms a stochastic PDE into a number of coupled deterministic PDEs. It has been successfully applied in several
engineering disciplines, for example structural mechanics, fluid dynamics and thermal engineering. Its applicability can further be enhanced by developing efficient numerical solution techniques for solving the coupled problems.
In this talk, we focus on solving high dimensional algebraic systems that result after discretizing steady-state linear stochastic PDEs. We shall present a thorough comparison of various iterative solvers. Multigrid approaches, preconditioned Krylov methods and block iterative splitting methods are considered. The multigrid
methods result in optimal convergence properties with respect to the stochastic and spatial discretization parameters. However, block splitting iterative methods
yield substantial lower computing times for a moderate stochastic accuracy and variance. Also, these latter methods reuse existing deterministic PDE solver routines. Numerical experiments shall be presented to illustrate the properties of the various methods.