1

July 2016
Journal of Computational Physics: Volume 316 Issue C, July 2016

**Publisher:** Academic Press Professional, Inc.

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 ...

**Keywords**:
Navier-Stokes equations, Uncertainty quantification, Stochastic Galerkin methods

2

March 2015
Journal of Computational Physics: Volume 284 Issue C, March 2015

**Publisher:** Academic Press Professional, Inc.

The magnetohydrodynamic (MHD) kinematics model describes the electromagnetic behavior of an electrically conducting fluid when its hydrodynamic properties are assumed to be known. In particular, the MHD kinematics equations can be used to simulate the magnetic field induced by a given velocity field. While prescribing the velocity field leads to ...

**Keywords**:
Kinematics equations, Magnetohydrodynamics, Iterative methods, Uncertainty quantification

3

March 2012
SIAM Journal on Scientific Computing: Volume 34 Issue 2, April 2012

**Publisher:** Society for Industrial and Applied Mathematics

We consider the numerical solution of a steady-state diffusion problem where the diffusion coefficient is the exponent of a random field. The standard stochastic Galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it. We consider a reformulated version of this ...

**Keywords**:
preconditioning, Karhunen-Loève expansion, finite elements, lognormal random field, algebraic multigrid, convection-diffusion problem, stochastic Galerkin method

4

Ray Tuminaro,

Michele Benzi,

Xiao-Chuan Cai,

Iain Duff,

Howard Elman,

Roland Freund,

Kirk Jordan,

Tim Kelley,

David Keyes,

Misha Kilmer,

Sven Leyffer,

Tom Manteuffel,

Steve McCormick,

David Silvester,

Homer Walker,

Carol Woodward,

Irad Yavneh
October 2011
SIAM Journal on Scientific Computing: Volume 33 Issue 5, September 2011

**Publisher:** Society for Industrial and Applied Mathematics

The biennial Copper Mountain Conference on Iterative Methods was held April 4-9, 2010. This meeting included more than 140 presentations covering many scientific computing areas, such as uncertainty quantification, optimization, Markov chains, saddle-point systems, inverse problems, direct factorizations, Krylov methods, algebraic multigrid, software frameworks, and advanced computer architectures. A partial ...

5

May 2011
Journal of Computational Physics: Volume 230 Issue 10, May, 2011

**Publisher:** Academic Press Professional, Inc.

We outline a new class of robust and efficient methods for solving the Navier-Stokes equations with a Boussinesq model for buoyancy driven flow. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for ...

**Keywords**:
Adaptivity, Algebraic multigrid, Boussinesq, Finite element approximation, Preconditioning, Navier-Stokes, Time stepping

6

January 2008
Journal of Computational Physics: Volume 227 Issue 3, January, 2008

**Publisher:** Academic Press Professional, Inc.

In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented ...

**Keywords**:
Incompressible flow, Iterative methods, Navier-Stokes

7

December 2007
SIAM Journal on Scientific Computing: Volume 30 Issue 1, November 2007

**Publisher:** Society for Industrial and Applied Mathematics

This paper introduces two stabilization schemes for the least squares commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth, and Tuminaro [ SIAM J. Sci. Comput. , 27 (2006), pp. 1651-1668] for the incompressible Navier-Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier-Stokes equations, ...

**Keywords**:
preconditioning, Navier-Stokes, iterative algorithms

8

June 2007
ACM Transactions on Mathematical Software (TOMS): Volume 33 Issue 2, June 2007

**Publisher:** ACM

**Bibliometrics**:

Citation Count: 46

Downloads (6 Weeks): 13, Downloads (12 Months): 113, Downloads (Overall): 1,733

Full text available:

PDF
IFISS is a graphical Matlab package for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions. The package can also be used as a computational laboratory for experimenting with state-of-the-art preconditioned iterative ...

**Keywords**:
finite elements, incompressible flow, stabilization, Matlab, iterative solvers

9

March 2007
Computing and Visualization in Science: Volume 10 Issue 1, March 2007

**Publisher:** Springer-Verlag

This paper is concerned with the convergence behaviour of multigrid methods for two- dimensional discrete convection-diffusion equations. In Elman and Ramage (BIT 46:283---299, 2006), we showed that for constant coefficient problems with grid-aligned flow and semiperiodic boundary conditions, the two-grid iteration matrix can be reduced via a set of orthogonal ...

10

February 2007
Computing and Visualization in Science: Volume 10 Issue 1, February 2007

**Publisher:** Springer-Verlag

This paper is concerned with the convergence behaviour of multigrid methods for two- dimensional discrete convection-diffusion equations. In Elman and Ramage (BIT 46:283–299, 2006), we showed that for constant coefficient problems with grid-aligned flow and semiperiodic boundary conditions, the two-grid iteration matrix can be reduced via a set of orthogonal ...

11

December 2006
SIAM Journal on Scientific Computing: Volume 28 Issue 6, December 2006

**Publisher:** Society for Industrial and Applied Mathematics

The discrete convection-diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes. Estimates of error reduction rates for both geometric multigrid (GMG) and algebraic multigrid (AMG) are established on uniform rectangular meshes for a model problem. Our analysis shows that GMG with ...

**Keywords**:
convection-diffusion equations, adaptive mesh refinement, multigrid

12

November 2005
SIAM Journal on Scientific Computing: Volume 27 Issue 5, 2006

**Publisher:** Society for Industrial and Applied Mathematics

This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible Navier--Stokes equations. We consider the "pressure convection--diffusion preconditioners" proposed by Kay, Loghin, and Wathen [ SIAM J. Sci. Comput. , 24 (2002), pp. 237-256] and Silvester, Elman, Kay, and Wathen [ J. Comput. Appl. Math. ...

**Keywords**:
Navier--Stokes, iterative algorithms, preconditioning

13

October 2005
Journal of Scientific Computing: Volume 25 Issue 1, October 2005

**Publisher:** Plenum Press

We describe some new preconditioning strategies for handling the algebraic systems of equations that arise from discretization of the incompressible Navier---Stokes equations. We demonstrate how these methods adapt in a straightforward manner to decisions on implicit or explicit time discretization, explore their use on a collection of benchmark problems, and ...

**Keywords**:
preconditioning, solvers, Navier---Stokes equations, incompressible fluids

14

May 2003
Journal of Computational Physics: Volume 187 Issue 2, 20 May 2003

**Publisher:** Academic Press Professional, Inc.

The development of robust and efficient algorithms for both steady-state simulations and fully implicit time integration of the Navier-Stokes equations is an active research topic. To be effective, the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence. In particular, they should be insensitive ...

**Keywords**:
Navier-Stokes, approximate block factorization, convention-diffusion operator, multi-level, approximate Schur complement, preconditioner, algebraic multigrid

15

January 2003
Mathematics of Computation: Volume 72 Issue 241, 01 January 2003

**Publisher:** American Mathematical Society

It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant ...

**Keywords**:
Galerkin finite element method, oscillations, convection-diffusion equation

16

October 2002
Applied Numerical Mathematics: Volume 43 Issue 1-2, October 2002

**Publisher:** Elsevier Science Publishers B. V.

Discretization and linearization of the incompressible Navier-Stokes equations leads to linear algebraic systems in which the coefficient matrix has the form of a saddle point problem ( F B B T O) (u p) = (f g ). In this paper, we describe the development of efficient and general iterative ...

17

January 2002
SIAM Journal on Numerical Analysis: Volume 40 Issue 1, 2002

**Publisher:** Society for Industrial and Applied Mathematics

Using a technique for constructing analytic expressions for discrete solutions to the convection-diffusion equation, we examine and characterize the effects of upwinding strategies on solution quality. In particular, for grid-aligned flow and discretization based on bilinear finite elements with streamline upwinding, we show precisely how the amount of upwinding included ...

**Keywords**:
convection-diffusion equation, Galerkin finite element method, oscillations, streamline diffusion

18

April 2001
SIAM Journal on Scientific Computing: Volume 23 Issue 4, 2001

**Publisher:** Society for Industrial and Applied Mathematics

Standard multigrid algorithms have proven ineffective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's effectiveness through theoretical analysis of a model problem and experimental results. In ...

**Keywords**:
multigrid, Helmholtz equation, Krylov subspace methods

19

March 2001
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations: Volume 128 Issue 1-2, March 2001

**Publisher:** Elsevier Science Publishers B. V.

**Keywords**:
multigrid iteration, preconditioning, Navier—Stokes equations, incompressible flow

20

December 1999
SIAM Journal on Scientific Computing: Volume 21 Issue 5, April 2000

**Publisher:** Society for Industrial and Applied Mathematics

This is the fourth in a series of special issues of SIAM Journal on Scientific Computing devoted to iterative methods for solving systems of algebraic equations. The results in these papers were originally presented at the Fifth Copper Mountain meeting on iterative methods, held in April 1998. There were 210 ...

**Keywords**:
local refinement, Navier--Stokes equations, irregular node, special linear quadrilateral finite elements