Generalized polynomial chaos decomposition and spectral methods for the stochastic Stokes equations

Abstract

In this paper we propose and analyze a high order numerical method to solve the Stokes equations with random coefficients. A stochastic Galerkin approach, based on a finite dimensional Karhunen–Loève decomposition technique for the stochastic inputs, is used to reduce the original stochastic Stokes equations into a set of deterministic equations for the expansion coefficients. Then a PN×PN-2 spectral method, together with a block Jacobi iteration is applied to solve the resulting problem. We establish the well-posedness of the weak formulation and its discrete counterpart. Moreover, we provide a rigorous convergence analysis and demonstrate exponential convergence with respect to the degrees of the polynomials chaos expansion used for the approximation in the random direction. Finally, a series of numerical tests is presented to support the theoretical results.