Multigrid for partial differential equations with stochastic coefficients

Mathematical models involving uncertain coefficients can often be modelled as stochastic partial differential equations. Typically, such equations are solved by Monte Carlo simulation. This is a robust and easily implementable technique. Unfortunately, its computational complexity rapidly becomes prohibitively expensive when large numbers of simulations are to be performed. As an alternative, a stochastic finite element method has been proposed. It extends the classical finite element method in a very natural way towards the stochastic dimension. In this method the stochastic PDE is transformed into a coupled system of deterministic PDEs. Classical space-time discretization methods can then be applied to convert this into a large set of algebraic equations.

In this work, a multigrid method is presented to solve the algebraic systems that result from stochastic finite element discretizations. The algorithm has very favorable convergence properties, such as a mesh-independent convergence rate. These properties are demonstrated through a local Fourier analysis, applied to the geometric multigrid variant. Also, the algebraic variant is discussed, and numerical results illustrate its convergence behavior. For time-dependent stochastic PDEs, the multigrid method is integrated with a high-order implicit Runge-Kutta time-integration scheme.