Algebraic multigrid for stationary and time-dependent PDEs with stochastic coefficients

Randomness in a physical problem can be modelled with probabilistic models such as stochastic partial differential equations (PDE). These equations contains some random parameters, for example, random coefficients in the differential operator or a random forcing term. To obtain all statistical information about the solution, the stochastic PDE can be solved using Monte Carlo simulations or using the stochastic finite element method. The latter method tries to reduce the computational cost of Monte Carlo simulations. It transforms a stochastic PDE into a coupled system of deterministic PDEs, that can further be discretized with deterministic finite element techniques. An algebraic multigrid method is presented to solve the algebraic systems resulting from stochastic finite element discretizations. Linear stationary and time-dependent stochastic PDEs are considered. The time discretization is performed by an implicit Runge-Kutta method. The numerical tests are based on a diffusion equation with a random diffusion coefficient that can be represented by Gaussian or uniform distributed random variables. The numerical tests and convergence analysis confirm optimal convergence properties with respect to the physical space discretization.status: publishe