Iterative Solvers for Stochastic Galerkin Discretizations of Stokes Flow with Random Data

In the field of uncertainty quantification, the effects of data uncertainties on the solution of a mathematical model are investigated. In order to quantify these effects, the stochastic Galerkin method can be applied. The underlying procedure relies on a representation of the solution in a generalized polynomial chaos basis and a subsequent Galerkin projection to compute the unknown coefficients. In comparison to stochastic sampling approaches such as the Monte Carlo method, stochastic Galerkin approaches exhibit better convergence rates, given the dependence of the solution on the stochastic input is smooth. The application of stochastic Galerkin discretizations does, however, involve the solution of large coupled systems of equations.

In this thesis, stochastic Galerkin discretizations of the Stokes equations with random data are investigated. Two different models for the uncertain viscosity are considered. The first model is an affine expansion with uniform random variables and the second one is a lognormal representation with Gaussian random variables. Variational formulations are derived based on the different input representations and well-posedness of the respective weak equations is shown. Subsequently, these equations are discretized using a stochastic Galerkin finite element approach. The spectral properties of the emerging systems of equations are investigated by the help of eigenvalue estimates.

Iterative approaches with suitable preconditioners are considered as solvers for the emerging large coupled systems of equations. A standard MINRES method with block diagonal preconditioner and a Bramble-Pasciak conjugate gradient method relying on a block triangular preconditioner are investigated. In order to establish bounds on the eigenvalues of the block diagonal and block triangular preconditioned system matrices, bounds on the eigenvalues of the respective relevant preconditioned sub-matrices are derived. These eigenvalue bounds are used to establish a connection between the spectral properties of the systems of equations under investigation and the convergence behavior of the iterative solvers.

The expected convergence behavior is illustrated using the flow in a driven cavity and the backward-facing step as numerical test cases. In the numerical experiments, a Bramble-Pasciak conjugate gradient method with a block triangular preconditioner converges faster than a standard MINRES method with a comparable block diagonal preconditioner.