Book•
Navier-Stokes Equations: Theory and Numerical Analysis
01 Jan 1979-
TL;DR: This paper presents thediscretization of the Navier-Stokes Equations: General Stability and Convergence Theorems, and describes the development of the Curl Operator and its application to the Steady-State Naviers' Equations.
Abstract: I. The Steady-State Stokes Equations . 1. Some Function Spaces. 2. Existence and Uniqueness for the Stokes Equations. 3. Discretization of the Stokes Equations (I). 4. Discretization of the Stokes Equations (II). 5. Numerical Algorithms. 6. The Penalty Method. II. The Steady-State Navier-Stokes Equations . 1. Existence and Uniqueness Theorems. 2. Discrete Inequalities and Compactness Theorems. 3. Approximation of the Stationary Navier-Stokes Equations. 4. Bifurcation Theory and Non-Uniqueness Results. III. The Evolution Navier-Stokes Equations . 1. The Linear Case. 2. Compactness Theorems. 3. Existence and Uniqueness Theorems. (n < 4). 4. Alternate Proof of Existence by Semi-Discretization. 5. Discretization of the Navier-Stokes Equations: General Stability and Convergence Theorems. 6. Discretization of the Navier-Stokes Equations: Application of the General Results. 7. Approximation of the Navier-Stokes Equations by the Projection Method. 8. Approximation of the Navier-Stokes Equations by the Artificial Compressibility Method. Appendix I: Properties of the Curl Operator and Application to the Steady-State Navier-Stokes Equations. Appendix II. (by F. Thomasset): Implementation of Non-Conforming Linear Finite Elements. Comments.
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TL;DR: In this paper, a numerical method for computing three-dimensional, time-dependent incompressible flows is presented based on a fractional-step, or time-splitting, scheme in conjunction with the approximate-factorization technique.
2,997 citations
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.
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.
2,253 citations
Additional excerpts
...Given a force field f : Ω → Rd and boundary data g : Γ → Rd, the problem is to find a velocity field u : Ω → Rd and a pressure field p : Ω → R such that −ν∆u + (u · ∇)u + ∇p = f in Ω, (2.1) ∇ · u = 0 in Ω, (2.2) Bu = g on Γ, (2.3) where ν > 0 is the kinematic viscosity coefficient (inversely…...
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01 Jan 2009
TL;DR: In this paper, a criterion for the convergence of numerical solutions of Navier-Stokes equations in two dimensions under steady conditions is given, which applies to all cases, of steady viscous flow in 2D.
Abstract: A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.
1,568 citations
Dissertation•
01 Jan 1996TL;DR: An automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy, based on a new stabilised and bounded second-order differencing scheme proposed.
Abstract: The accuracy of numerical simulation algorithms is one of main concerns in modern Computational Fluid Dynamics. Development of new and more accurate mathematical models requires an insight into the problem of numerical errors. In order to construct an estimate of the solution error in Finite Volume calculations, it is first necessary to examine its sources. Discretisation errors can be divided into two groups: errors caused by the discretisation of the solution domain and equation discretisation errors. The first group includes insufficient mesh resolution, mesh skewness and non-orthogonality. In the case of the second order Finite Volume method, equation discretisation errors are represented through numerical diffusion. Numerical diffusion coefficients from the discretisation of the convection term and the temporal derivative are derived. In an attempt to reduce numerical diffusion from the convection term, a new stabilised and bounded second-order differencing scheme is proposed. Three new methods of error estimation are presented. The Direct Taylor Series Error estimate is based on the Taylor series truncation error analysis. It is set up to enable single-mesh single-run error estimation. The Moment Error estimate derives the solution error from the cell imbalance in higher moments of the solution. A suitable normalisation is used to estimate the error magnitude. The Residual Error estimate is based on the local inconsistency between face interpolation and volume integration. Extensions of the method to transient flows and the Local Residual Problem error estimate are also given. Finally, an automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy. It uses mesh refinement and unrefinement to control the local error magnitude. The method is tested on several characteristic flow situations, ranging from incompressible to supersonic flows, for both steady-state and transient problems.
1,418 citations
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.
1,230 citations
Cites background from "Navier-Stokes Equations: Theory and..."
...1 The work of this author has been supported by CNRS and ICES at UT under ICES Visiting Faculty Fellowships....
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81 citations
01 Jan 1976
TL;DR: In this article, several regularity results for the Stokes problem in a polygonal domain have been discussed, based on the method used by Kondratev to study the regularity of a single 2mth order elliptic equation.
Abstract: Publisher Summary This chapter discusses several regularity results for the Stokes problem in a polygonal domain. Analogous regularity results for solutions of a single 2mth order elliptic equation in a polygonal domain have been extensively developed. Regularity results are of fundamental importance in the analysis of numerical methods for the Stokes problem. Also, regularity results are used in analyzing the stability of stationary solutions of the Navier-Stokes equations. The results presented in the chapter are all based on the method used by Kondratev to study the regularity of a single 2mth order elliptic equation. The results depend in an essential way on the spectral properties of a system of ordinary differential equations, which is associated with the Stokes equations; the chapter discusses this dependence. The chapter also discusses two low order regularity results and two higher order results.
14 citations