A Bramble-Pasciak conjugate gradient method for discrete Stokes problems with lognormal random viscosity
For researchers solving stochastic saddle point problems, this work introduces a more efficient iterative solver alternative, though the improvement is demonstrated only on a single numerical example.
The paper studies iterative solvers for Stokes flow with random viscosity, showing that the Bramble-Pasciak conjugate gradient method with a block triangular preconditioner outperforms the MINRES method with a block diagonal preconditioner in terms of iteration counts for a cavity flow test case.
We study linear systems of equations arising from a stochastic Galerkin finite element discretization of saddle point problems with random data and its iterative solution. We consider the Stokes flow model with random viscosity described by the exponential of a correlated random process and shortly discuss the discretization framework and the representation of the emerging matrix equation. Due to the high dimensionality and the coupling of the associated symmetric, indefinite, linear system, we resort to iterative solvers and problem-specific preconditioners. As a standard iterative solver for this problem class, we consider the block diagonal preconditioned MINRES method and further introduce the Bramble-Pasciak conjugate gradient method as a promising alternative. This special conjugate gradient method is formulated in a non-standard inner product with a block triangular preconditioner. From a structural point of view, such a block triangular preconditioner enables a better approximation of the original problem than the block diagonal one. We derive eigenvalue estimates to assess the convergence behavior of the two solvers with respect to relevant physical and numerical parameters and verify our findings by the help of a numerical test case. We model Stokes flow in a cavity driven by a moving lid and describe the viscosity by the exponential of a truncated Karhunen-Loeve expansion. Regarding iteration counts, the Bramble-Pasciak conjugate gradient method with block triangular preconditioner is superior to the MINRES method with block diagonal preconditioner in the considered example.