Stochastic Galerkin methods for the steady-state Navier-Stokes equations

TL;DR

This paper develops stochastic Galerkin methods to solve steady-state Navier-Stokes equations with random viscosity, proposing efficient linearization and preconditioning techniques, and demonstrating their effectiveness on benchmark problems.

Contribution

It introduces a stochastic Galerkin framework for Navier-Stokes with random viscosity, including new linearization schemes and a preconditioner for improved computational efficiency.

Findings

Effective preconditioner for stochastic linear systems

Successful application to benchmark problems

Enhanced understanding of stochastic solution properties

Abstract

We study the steady-state Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems.