Algebraic multigrid for stationary and time-dependent partial differential equations with stochastic coefficients

We consider the numerical solution of time-dependent partial differential equations with random coefficients. A spectral approach, called stochastic finite element method, is used to compute the statistical characteristics of the solution. This method transforms a stochastic partial differential equation into a coupled system of deterministic equations by means of a Galerkin projection onto a generalized polynomial chaos. An algebraic multigrid method is presented to solve the algebraic systems that result after discretization of this coupled system. High-order time integration schemes of an implicit Runge-Kutta type, and spatial discretization on unstructured finite element meshes are considered. The convergence properties of the algebraic multigrid method are demonstrated by a convergence analysis and by numerical tests.