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
13 - The Finite Element Method for Three Dimensional Incompressible Flow
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
The finite element method has been an established method for approximating incompressible flow for a number of years. The primitive variable, velocity/pressure formulation, is the most popular way to implement the method although it is not the only possibility. Much theoretical work has been done to establish convergence and error estimates and there is a large amount of literature on the topic, see for example Temam [1984], Thomasset [1981], Girault and Raviart [1986], Gunzburger [1989]. Effort has been concentrated on two-dimensional flow and although mathematically three-dimensional flow is no more difficult, in practice the current state of computer hardware makes the implementation of three-dimensional elements much more problematic. It is only the availability of modern supercomputers that has allowed the approximation of such flows to be attempted. However, an element and method of solution that works well on a large vector processor may be quite inefficient on a fine-grained parallel computer thus the concept of the “best element” or even a “good element” may be highly dependent on the computer on which it is to be implemented.
Most methods of solution involve at least one iteration of one form or another, the innermost loop being the solution of a set of linear equations. For practical three dimensional flow problems a direct method of solution is unlikely to be a feasible proposition for almost all situations and this inner system of equations will have to be solved iteratively. “How accurately do we need to solve this system?” and “What are the interactions between the various iterations taking place?” are two questions that have to be faced by anyone implementing the finite element method for three-dimensional flow.
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- Information
- Incompressible Computational Fluid DynamicsTrends and Advances, pp. 427 - 446Publisher: Cambridge University PressPrint publication year: 1993
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