Finite element methods for incompressible viscous flow

doi:10.1016/S1570-8659(03)09003-3

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Finite element methods for incompressible viscous flow

Book Chapter•DOI•

Finite element methods for incompressible viscous flow

Roland Glowinski¹, Roland Glowinski²•Institutions (2)

Pierre-and-Marie-Curie University¹, University of Houston²

01 Jan 2003-Handbook of Numerical Analysis (Elsevier)-Vol. 9, pp 3-1176

About: This article is published in Handbook of Numerical Analysis.The article was published on 2003-01-01. It has received 455 citations till now. The article focuses on the topics: Extended finite element method & Pressure-correction method.

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Journal Article•DOI•

Numerical solution of saddle point problems

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Michele Benzi¹, Gene H. Golub², Jörg Liesen³•Institutions (3)

Emory University¹, Stanford University², Technical University of Berlin³

01 May 2005-Acta Numerica

TL;DR: A large selection of solution methods for linear systems in saddle point form are presented, with an emphasis on iterative methods for large and sparse problems.

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Abstract: Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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2,253 citations

Cites methods from "Finite element methods for incompre..."

...They also arise as subproblems in the numerical solution of the Navier–Stokes equations by operator splitting methods (Glowinski 2003, Quarteroni and Valli 1994) and as the first step of Picard’s iteration when the initial guess used is u(0) = 0....
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...The literature on numerical methods for incompressible flow problems is vast; see, e.g., Elman et al. (2005c), Fortin (1993), Glowinski (2003), Gresho and Sani (1998), Gunzburger (1989), Quarteroni and Valli (1994), Temam (1984), Turek (1999) and Wesseling (2001)....
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...We start with Uzawa’s method (Uzawa 1958), which enjoys considerable popularity in fluid dynamics, especially for solving the (steady) Stokes problem (Fortin and Glowinski 1983, Glowinski 1984, Glowinski 2003, Temam 1984, Turek 1999)....
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Journal Article•DOI•

An overview of projection methods for incompressible flows

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Jean-Luc Guermond¹, Jean-Luc Guermond², Peter D. Minev³, Jie Shen⁴•Institutions (4)

Texas A&M University¹, Centre national de la recherche scientifique², University of Alberta³, Purdue University⁴

15 Sep 2006-Computer Methods in Applied Mechanics and Engineering

TL;DR: In this paper, a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed, and the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners.

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1,230 citations

Journal Article•DOI•

A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow

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Roland Glowinski¹, Tsorng-Whay Pan¹, T. I. Helsa², Daniel D. Joseph², Jacques Periaux³ - Show less +1 more•Institutions (3)

University of Houston¹, University of Minnesota², Dassault Aviation³

30 May 2001-Journal of Computational Physics

TL;DR: In this paper, the Lagrange-multiplier-based fictitious domain methods are combined with finite element approximations of the Navier-Stokes equations occurring in the global model to simulate incompressible viscous fluid flow past moving rigid bodies.

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982 citations

Cites methods from "Finite element methods for incompre..."

...[37, 43], while the various elliptic problems involved in our methodology have been treated by fast elliptic solvers based on cyclic reduction....
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...• If the least-squares/conjugate gradient methodology advocated in [37] and [43] is used to treat the advection–diffusion it requires two (preconditioned) iterations at most....
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Spectral and High-Order Methods with Applications

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Jie Shen, Tao Tang, Zhen-Huan Teng

01 Jan 2006

415 citations

Cites methods from "Finite element methods for incompre..."

...We refer to the books[162, 89, 40, 56] for more details on the approximation of Navier-Stokes equations by finite elements, spectral and spectral element methods....
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Journal Article•DOI•

DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition

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Vivette Girault, Béatrice Rivière¹•Institutions (1)

Rice University¹

01 Apr 2009-SIAM Journal on Numerical Analysis

TL;DR: This work couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffman's condition on the interface, to prove existence of a weak solution as well as some a priori estimates.

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Abstract: In this work, we couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffman's condition on the interface. Under suitable smallness conditions on the data, we prove existence of a weak solution as well as some a priori estimates. We establish local uniqueness when the data satisfy additional smallness restrictions. Then we propose a discontinuous Galerkin scheme for discretizing the equations and do its numerical analysis.

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223 citations

Cites methods from "Finite element methods for incompre..."

...Most algorithms for solving numerically the discrete Navier–Stokes equations (see, for instance, [23]) can be applied to solve the nonlinear scheme (5....
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References

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Book•

Matrix computations

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Gene H. Golub

01 Jan 1983

34,729 citations

Book•

The finite element method

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O. C. Zienkiewicz

01 Jan 1989

TL;DR: In this article, the methodes are numeriques and the fonction de forme reference record created on 2005-11-18, modified on 2016-08-08.

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Abstract: Keywords: methodes : numeriques ; fonction de forme Reference Record created on 2005-11-18, modified on 2016-08-08

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17,327 citations

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Iterative Methods for Sparse Linear Systems

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Yousef Saad

01 Apr 2003

TL;DR: This chapter discusses methods related to the normal equations of linear algebra, and some of the techniques used in this chapter were derived from previous chapters of this book.

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Abstract: Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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13,484 citations

Book•

The Finite Element Method for Elliptic Problems

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Philippe G. Ciarlet, J. T. Oden¹•Institutions (1)

University of Texas at Austin¹

01 Jan 1978

TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.

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Abstract: Preface 1. Elliptic boundary value problems 2. Introduction to the finite element method 3. Conforming finite element methods for second-order problems 4. Other finite element methods for second-order problems 5. Application of the finite element method to some nonlinear problems 6. Finite element methods for the plate problem 7. A mixed finite element method 8. Finite element methods for shells Epilogue Bibliography Glossary of symbols Index.

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8,407 citations

Book•

Finite Element Method for Elliptic Problems

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Philippe G. Ciarlet

01 Apr 2002

TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.

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Abstract: From the Publisher: This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained. About the Author Philippe G. Ciarlet is a Professor at the Laboratoire d'Analyse Numerique at the Universite Pierre et Marie Curie in Paris. He is also a member of the French Academy of Sciences. He is the author of more than a dozen books on a variety of topics and is a frequent invited lecturer at meetings and universities throughout the world. Professor Ciarlet has served approximately 75 visiting professorships since 1973, and he is a member of the editorial boards of more than 20 journals.

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8,052 citations