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

## 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.

- Authors:

- Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 (United States)
- Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742 (United States)

- Publication Date:

- OSTI Identifier:
- 22572324

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 316; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BENCHMARKS; CHAOS THEORY; FINITE ELEMENT METHOD; NAVIER-STOKES EQUATIONS; NONLINEAR PROBLEMS; POLYNOMIALS; RANDOMNESS; STEADY-STATE CONDITIONS; STOCHASTIC PROCESSES

### Citation Formats

```
Sousedík, Bedřich, E-mail: sousedik@umbc.edu, and Elman, Howard C., E-mail: elman@cs.umd.edu.
```*Stochastic Galerkin methods for the steady-state Navier–Stokes equations*. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.04.013.

```
Sousedík, Bedřich, E-mail: sousedik@umbc.edu, & Elman, Howard C., E-mail: elman@cs.umd.edu.
```*Stochastic Galerkin methods for the steady-state Navier–Stokes equations*. United States. doi:10.1016/J.JCP.2016.04.013.

```
Sousedík, Bedřich, E-mail: sousedik@umbc.edu, and Elman, Howard C., E-mail: elman@cs.umd.edu. Fri .
"Stochastic Galerkin methods for the steady-state Navier–Stokes equations". United States.
doi:10.1016/J.JCP.2016.04.013.
```

```
@article{osti_22572324,
```

title = {Stochastic Galerkin methods for the steady-state Navier–Stokes equations},

author = {Sousedík, Bedřich, E-mail: sousedik@umbc.edu and Elman, Howard C., E-mail: elman@cs.umd.edu},

abstractNote = {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.},

doi = {10.1016/J.JCP.2016.04.013},

journal = {Journal of Computational Physics},

number = ,

volume = 316,

place = {United States},

year = {Fri Jul 01 00:00:00 EDT 2016},

month = {Fri Jul 01 00:00:00 EDT 2016}

}

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