Research Topic

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. It 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 a stochastic Galerkin discretization does, however, involve the solution of large coupled systems of equations.
In this project, 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 corresponding 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 based on eigenvalue analysis.
Iterative methods with suitable preconditioners are considered as solvers for the large coupled systems of equations. A block diagonal preconditioned MINRES method and a Bramble-Pasciak conjugate gradient method relying on a block triangular preconditioner are investigated. It is shown that such a special conjugate gradient method exists in the setting of this project. Bounds on the eigenvalues of the relevant preconditioned sub-matrices are derived in order to establish bounds on the eigenvalues of the block diagonal and block triangular preconditioned system matrices. These eigenvalue estimates are used to establish a connection between the spectral properties of the systems of equations under investigation and the convergence behavior of the utilized iterative methods.
The expected convergence behavior is illustrated using the flow in a driven cavity and the backward facing step as numerical test cases. The corresponding uncertain viscosity is modeled using two different covariance functions and the associated Karhunen-Loève expansions. In the numerical experiments, a Bramble-Pasciak conjugate gradient method with block triangular preconditioner outperforms a MINRES method with block diagonal preconditioner in terms of computational complexity.

Key Research Area

Uncertainty quantification, stochastic Galerkin method, fluid dynamics, iterative solvers, preconditioning

Contact

Christopher Müller,
M.Sc.

Address:

Dolivostraße 15

D-64293 Darmstadt

Germany

Phone:

+49 6151 16 - 24395

Fax:

+49 6151 16 - 24404

Office:

S4|10-227

Email:

cmueller (at) gsc.tu...

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