Iterative solvers for the stochastic finite element method

The need for accurate simulations and reliability estimates of predictions has led to a variety of techniques to mathematically model and quantify uncertainty and variability. Depending on the problem characteristics, techniques based on probability theory or non-probabilistic techniques, for example based on fuzzy theory, are used. In the first category, the stochastic finite element method has received a lot of attention in recent years. This method transforms a stochastic partial differential equation (PDE) into a large coupled system of deterministic PDEs. It has been successfully applied to 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 the resulting discretized systems.In this work, we will focus on multigrid solvers for the algebraic systems that result from stochastic finite element discretizations. As the stochastic finite element method increases the dimension of the original problem, specialized large-scale solvers are required. Multigrid methods are widely used to solve large-scale algebraic systems that result from discretized PDEs. We have developed a multigrid solution method for stochastic Galerkin finite element discretizations. The method is applicable to stationary and time-dependent problems, possibly combined with a high-order time discretization scheme. In addition, we constructed a multigrid approach that combines a hierarchy in the spatial and the stochastic dimension. We will point out the favorable convergence properties of our developed solvers and compare them to the state-of-art.

ISBN:9789073802858

Publication year:2008