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

Sousedík, Bedřich; Elman, Howard C.

doi:10.1016/j.jcp.2016.04.013

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

Journal Article · Tue Apr 12 00:00:00 EDT 2016 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2016.04.013· OSTI ID:1418535

^[1]; Elman, Howard C. ^[2]

Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States); DOE Office of Scientific and Technical Information (OSTI)
Univ. of Maryland, College Park, MD (United States)

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.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Maryland, College Park, MD (United States)

Sponsoring Organization:: National Science Foundation; USDOE

Grant/Contract Number:: SC0009301

OSTI ID:: 1418535

Alternate ID(s):: OSTI ID: 22572324
OSTI ID: 1467148

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 316; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (21)

Convexity and Solvability for Compactly Supported Radial Basis Functions with Different Shapes Zhu, Shengxin; Wathen, Andrew J. Journal of Scientific Computing, Vol. 63, Issue 3 https://doi.org/10.1007/s10915-014-9919-9	journal	October 2014
Iterative Solvers for the Stochastic Finite Element Method Rosseel, Eveline; Vandewalle, Stefan SIAM Journal on Scientific Computing, Vol. 32, Issue 1 https://doi.org/10.1137/080727026	journal	January 2010
Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods: Dedicated to Professor Ivo Marek on the occasion of his 80th birthday. Sousedík, Bedřich; Ghanem, Roger G.; Phipps, Eric T. Numerical Linear Algebra with Applications, Vol. 21, Issue 1 https://doi.org/10.1002/nla.1869	journal	January 2013
Smoothed aggregation algebraic multigrid for stochastic PDE problems with layered materials Brezina, Marian; Doostan, Alireza; Manteuffel, Tom Numerical Linear Algebra with Applications, Vol. 21, Issue 2 https://doi.org/10.1002/nla.1924	journal	February 2014
High dimensional integration of smooth functions over cubes Novak, Erich; Ritter, Klaus Numerische Mathematik, Vol. 75, Issue 1 https://doi.org/10.1007/s002110050231	journal	November 1996
Iterative solution of systems of linear equations arising in the context of stochastic finite elements Pellissetti, M. F.; Ghanem, R. G. Advances in Engineering Software, Vol. 31, Issue 8-9 https://doi.org/10.1016/S0965-9978(00)00034-X	journal	August 2000
Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations Matthies, Hermann G.; Keese, Andreas Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 12-16 https://doi.org/10.1016/j.cma.2004.05.027	journal	April 2005
Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics Elman, Howard; Silvester, David; Wathen, Andy https://doi.org/10.1093/acprof:oso/9780199678792.001.0001	book	May 2014
Block-diagonal preconditioning for spectral stochastic finite-element systems Powell, C. E.; Elman, H. C. IMA Journal of Numerical Analysis, Vol. 29, Issue 2 https://doi.org/10.1093/imanum/drn014	journal	April 2008
The Nonlinear Gaussian Spectrum of Log-Normal Stochastic Processes and Variables Ghanem, R. Journal of Applied Mechanics, Vol. 66, Issue 4 https://doi.org/10.1115/1.2791806	journal	December 1999
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data Babuška, Ivo; Nobile, Fabio; Tempone, Raúl SIAM Journal on Numerical Analysis, Vol. 45, Issue 3 https://doi.org/10.1137/050645142	journal	January 2007
Stochastic Galerkin Matrices Ernst, Oliver G.; Ullmann, Elisabeth SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 4 https://doi.org/10.1137/080742282	journal	January 2010
A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations Ullmann, Elisabeth SIAM Journal on Scientific Computing, Vol. 32, Issue 2 https://doi.org/10.1137/080742853	journal	January 2010
A Flexible Inner-Outer Preconditioned GMRES Algorithm Saad, Youcef SIAM Journal on Scientific Computing, Vol. 14, Issue 2 https://doi.org/10.1137/0914028	journal	March 1993
Preconditioning Steady-State Navier--Stokes Equations with Random Data Powell, Catherine E.; Silvester, David J. SIAM Journal on Scientific Computing, Vol. 34, Issue 5 https://doi.org/10.1137/120870578	journal	January 2012
Model Reduction Based on Proper Generalized Decomposition for the Stochastic Steady Incompressible Navier--Stokes Equations Tamellini, L.; Le Maître, O.; Nouy, A. SIAM Journal on Scientific Computing, Vol. 36, Issue 3 https://doi.org/10.1137/120878999	journal	January 2014
Reduced Basis Collocation Methods for Partial Differential Equations with Random Coefficients Elman, Howard C.; Liao, Qifeng SIAM/ASA Journal on Uncertainty Quantification, Vol. 1, Issue 1 https://doi.org/10.1137/120881841	journal	January 2013
IFISS: A Computational Laboratory for Investigating Incompressible Flow Problems Elman, Howard C.; Ramage, Alison; Silvester, David J. SIAM Review, Vol. 56, Issue 2 https://doi.org/10.1137/120891393	journal	January 2014
Low-Rank Approximate Inverse for Preconditioning Tensor-Structured Linear Systems Giraldi, L.; Nouy, A.; Legrain, G. SIAM Journal on Scientific Computing, Vol. 36, Issue 4 https://doi.org/10.1137/130918137	journal	January 2014
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations Xiu, Dongbin; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 24, Issue 2 https://doi.org/10.1137/S1064827501387826	journal	January 2002
Truncated Hierarchical Preconditioning for the Stochastic Galerkin fem Sousedik, Bedrich; Ghanem, Roger International Journal for Uncertainty Quantification, Vol. 4, Issue 4 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2014007353	journal	January 2014

Cited By (1)

An efficient algorithm for simulating ensembles of parameterized flow problems Gunzburger, Max; Jiang, Nan; Wang, Zhu IMA Journal of Numerical Analysis, Vol. 39, Issue 3 https://doi.org/10.1093/imanum/dry029	journal	May 2018

Similar Records

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

Journal Article · Fri Jul 01 00:00:00 EDT 2016 · Journal of Computational Physics · OSTI ID:22572324

A low-rank solver for the stochastic unsteady Navier–Stokes problem

Journal Article · Thu Mar 05 19:00:00 EST 2020 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:1801900

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Navier–Stokes equations
Stochastic Galerkin methods
Uncertainty quantification

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

Citation Formats

References (21)

Cited By (1)

Similar Records

Related Subjects