Book contents
- Frontmatter
- Contents
- Preface
- Contributing Authors
- 1 A Few Tools For Turbulence Models In Navier-Stokes Equations
- 2 On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow
- 3 CFD - An Industrial Perspective
- 4 Stabilized Finite Element Methods
- 5 Optimal Control and Optimization of Viscous, Incompressible Flows
- 6 A Fully-Coupled Finite Element Algorithm, Using Direct and Iterative Solvers, for the Incompressible Navier-Stokes Equations
- 7 Numerical Solution of the Incompressible Navier-Stokes Equations in Primitive Variables on Unstaggered Grids
- 8 Spectral Element and Lattice Gas Methods for Incompressible Fluid Dynamics
- 9 Design of Incompressible Flow Solvers: Practical Aspects
- 10 The Covolume Approach to Computing Incompressible Flows
- 11 Vortex Methods: An Introduction and Survey of Selected Research Topics
- 12 New Emerging Methods in Numerical Analysis: Applications to Fluid Mechanics
- 13 The Finite Element Method for Three Dimensional Incompressible Flow
- 14 A Posteriori Error Estimators and Adaptive Mesh-Refinement Techniques for the Navier-Stokes Equations
- Index
4 - Stabilized Finite Element Methods
Published online by Cambridge University Press: 12 January 2010
- Frontmatter
- Contents
- Preface
- Contributing Authors
- 1 A Few Tools For Turbulence Models In Navier-Stokes Equations
- 2 On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow
- 3 CFD - An Industrial Perspective
- 4 Stabilized Finite Element Methods
- 5 Optimal Control and Optimization of Viscous, Incompressible Flows
- 6 A Fully-Coupled Finite Element Algorithm, Using Direct and Iterative Solvers, for the Incompressible Navier-Stokes Equations
- 7 Numerical Solution of the Incompressible Navier-Stokes Equations in Primitive Variables on Unstaggered Grids
- 8 Spectral Element and Lattice Gas Methods for Incompressible Fluid Dynamics
- 9 Design of Incompressible Flow Solvers: Practical Aspects
- 10 The Covolume Approach to Computing Incompressible Flows
- 11 Vortex Methods: An Introduction and Survey of Selected Research Topics
- 12 New Emerging Methods in Numerical Analysis: Applications to Fluid Mechanics
- 13 The Finite Element Method for Three Dimensional Incompressible Flow
- 14 A Posteriori Error Estimators and Adaptive Mesh-Refinement Techniques for the Navier-Stokes Equations
- Index
Summary
Introduction
Physically the Stokes equations model “slow” flows of incompressible fluids or alternatively isotropic incompressible elastic materials. In Computational Fluid Dynamics, however, the Stokes equations have become an important model problem for designing and analyzing finite element algorithms. The reason being, that some of the problems encountered when solving the full Navier-Stokes equations are already present in the more simple Stokes equations. In particular, it gives the right setting for studying the stability problem connected with the choice of finite element spaces for the velocity and the pressure. It is well known that these spaces cannot be chosen independently when the discretization is based on the “Galerkin” variational form. This method belongs to the class of saddle-point problems for which an abstract theory has been developed by Brezzi [1974] and Babuska [1973]. The theory shows that the method is optimally convergent if the finite element spaces for velocity and pressure satisfy the “Babuska-Brezzi” or “inf-sup” condition. In computations the violation of this condition often leads to unphysical pressure oscillations and a “locking” of the velocity field, cf. Hughes [1987]. During the last decade this problem has been studied thoroughly and various velocity-pressure combinations have been shown to satisfy the Babuska-Brezzi condition. Unfortunately, however, it has turned out that many seemingly natural combinations do not satisfy it. (See Girault and Raviart [1986], Brezzi and Fortin [1991], and references therein.)
In this chapter we will review a recent technique of “stabilizing” mixed methods. In this approach the standard Galerkin form is modified by the addition of mesh-dependent terms which are weighted residuals of the differential equations.
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- Chapter
- Information
- Incompressible Computational Fluid DynamicsTrends and Advances, pp. 87 - 108Publisher: Cambridge University PressPrint publication year: 1993
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