Book•
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
19 Jun 1986-
TL;DR: This paper presents the results of an analysis of the "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions and its applications to Mixed Approximation and Homogeneous Stokes Equations.
Abstract: I. Mathematical Foundation of the Stokes Problem.- 1. Generalities on Some Elliptic Boundary Value Problems.- 1.1. Basic Concepts on Sobolev Spaces.- 1.2. Abstract Elliptic Theory.- 1.3. Example 1: Dirichlet's Problem for the Laplace Operator.- 1.4. Example 2: Neumann's Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- s Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- 2. Theory of the Steady-State Navier-Stokes Equations.- 2.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 2.2. The Stream Function Formulation of the Homogeneous Problem..- 3. Approximation of Branches of Nonsingular Solutions.- 3.1. An Abstract Framework.- 3.2. Approximation of Branches of Nonsingular Solutions.- 3.3. Application to a Class of Nonlinear Problems.- 3.4. Non-Differentiable Approximation of Branches of Nonsingular Solutions.- 4. Numerical Analysis of Centered Finite Element Schemes.- 4.1. Formulation in Primitive Variables: Methods Using Discontinuous Pressures.- 4.2. Formulation in Primitive Variables: the Case of Continuous Pressures.- 4.3. Mixed Incompressible Methods: the "Stream Function-Vorticity" Formulation.- 4.4. Remarks on the "Stream Function-Gradient of Velocity Tensor" Scheme.- 5. Numerical Analysis of Upwind Schemes.- 5.1. Upwinding in the Stream Function-Vorticity Scheme.- 5.2. Error Analysis of the Upwind Scheme.- 5.3. Approximating the Pressure with the Upwind Scheme.- 6. Numerical Algorithms.- 2.11. General Methods of Descent and Application to Gradient Methods.- 2.12. Least-Squares and Gradient Methods to Solve the Navier-Stokes Equations.- 2.13. Newton's Method and the Continuation Method.- References.- Index of Mathematical Symbols.
Citations
More filters
Book•
01 Jan 2000TL;DR: In this paper, a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics is presented, focusing on methods for linear elliptic boundary value problems.
Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.
2,607 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
1,097 citations
TL;DR: (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations are considered.
Abstract: In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.
1,090 citations
TL;DR: Finite element exterior calculus as mentioned in this paper is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations, which brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological and algebraic structures which underlie well-posedness of the PDE problem being solved.
Abstract: Finite element exterior calculus is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are re- vealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Lapla- cian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.
1,044 citations