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We study linear systems of equations arising from a stochastic Galerkinfinite element discretization of saddle point problems with random data and itsiterative solution. We consider the Stokes flow model with random viscositydescribed by the exponential of a correlated random process and shortly discussthe discretization framework and the representation of the emerging matrixequation. Due to the high dimensionality and the coupling of the associatedsymmetric, indefinite, linear system, we resort to iterative solvers andproblem-specific preconditioners. As a standard iterative solver for thisproblem class, we consider the block diagonal preconditioned MINRES method andfurther introduce the Bramble-Pasciak conjugate gradient method as a promisingalternative. This special conjugate gradient method is formulated in anon-standard inner product with a block triangular preconditioner. From astructural point of view, such a block triangular preconditioner enables abetter approximation of the original problem than the block diagonal one. Wederive eigenvalue estimates to assess the convergence behavior of the twosolvers with respect to relevant physical and numerical parameters and verifyour findings by the help of a numerical test case. We model Stokes flow in acavity driven by a moving lid and describe the viscosity by the exponential ofa truncated Karhunen-Loeve expansion. Regarding iteration counts, theBramble-Pasciak conjugate gradient method with block triangular preconditioneris superior to the MINRES method with block diagonal preconditioner in theconsidered example.

A Bramble-Pasciak conjugate gradient method for discrete Stokes problems with lognormal random viscosity