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

Viac o knihe

In uncertainty quantification, the effects of data uncertainties on mathematical model solutions are explored. The stochastic Galerkin method is employed to quantify these effects, relying on a generalized polynomial chaos basis and Galerkin projection to compute unknown coefficients. Compared to stochastic sampling methods like Monte Carlo, stochastic Galerkin approaches offer improved convergence rates when the solution's dependence on stochastic input is smooth. However, applying stochastic Galerkin discretizations necessitates solving large coupled systems of equations. This research investigates stochastic Galerkin discretizations of the Stokes equations with random data, focusing on two uncertain viscosity models: an affine expansion with uniform random variables and a lognormal representation with Gaussian random variables. Variational formulations are developed for these input representations, demonstrating well-posedness of the respective weak equations. These equations are then discretized using a stochastic Galerkin finite element approach. The spectral properties of the resulting systems are analyzed through eigenvalue estimates, and iterative solvers with suitable preconditioners are considered. The study evaluates a standard MINRES method with a block diagonal preconditioner and a Bramble-Pasciak conjugate gradient method with a block triangular preconditioner. Eigenvalue bounds are established for the preconditi