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

TL;DR

This paper introduces a Bramble-Pasciak conjugate gradient method with a block triangular preconditioner for solving stochastic Stokes equations with random viscosity, demonstrating improved efficiency over traditional methods.

Contribution

The paper develops a novel Bramble-Pasciak conjugate gradient method tailored for stochastic saddle point problems with block triangular preconditioning, enhancing solver performance.

Findings

The new method outperforms MINRES in iteration count.

Eigenvalue bounds are derived for preconditioned matrices.

Numerical tests confirm improved efficiency in flow with random viscosity.

Abstract

We study the iterative solution of linear systems of equations arising from stochastic Galerkin finite element discretizations of saddle point problems. We focus on the Stokes model with random data parametrized by uniformly distributed random variables and discuss well-posedness of the variational formulations. We introduce a Bramble-Pasciak conjugate gradient method as a linear solver. It builds on a non-standard inner product associated with a block triangular preconditioner. The block triangular structure enables more sophisticated preconditioners than the block diagonal structure usually applied in MINRES methods. We show how the existence requirements of a conjugate gradient method can be met in our setting. We analyze the performance of the solvers depending on relevant physical and numerical parameters by means of eigenvalue estimates. For this purpose, we derive bounds for the eigenvalues of the relevant preconditioned sub-matrices. We illustrate our findings using the flow in a driven cavity as a numerical test case, where the viscosity is given by a truncated Karhunen-Lo\`eve expansion of a random field. In this example, a Bramble-Pasciak conjugate gradient method with block triangular preconditioner outperforms a MINRES method with block diagonal preconditioner in terms of iteration numbers.