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Preface |
7 |
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Contents |
8 |
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1 Introduction |
13 |
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1.1 Contents of this Monograph |
14 |
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2 The Navier–Stokes Equations as Model for Incompressible Flows |
19 |
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2.1 The Conservation of Mass |
19 |
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2.2 The Conservation of Linear Momentum |
21 |
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2.3 The Dimensionless Navier–Stokes Equations |
29 |
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2.4 Initial and Boundary Conditions |
31 |
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3 Finite Element Spaces for Linear Saddle Point Problems |
37 |
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3.1 Existence and Uniqueness of a Solution of an Abstract Linear Saddle Point Problem |
38 |
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3.2 Appropriate Function Spaces for Continuous Incompressible Flow Problems |
53 |
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3.3 General Considerations on Appropriate Function Spaces for Finite Element Discretizations |
64 |
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3.4 Examples of Pairs of Finite Element Spaces Violating the Discrete Inf-Sup Condition |
74 |
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3.5 Techniques for Checking the Discrete Inf-Sup Condition |
84 |
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3.5.1 The Fortin Operator |
84 |
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3.5.2 Splitting the Discrete Pressure into a Piecewise Constant Part and a Remainder |
88 |
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3.5.3 An Approach for Conforming Velocity Spaces and Continuous Pressure Spaces |
91 |
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3.5.4 Macroelement Techniques |
96 |
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3.6 Inf-Sup Stable Pairs of Finite Element Spaces |
105 |
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3.6.1 The MINI Element |
105 |
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3.6.2 The Family of Taylor–Hood Finite Elements |
110 |
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3.6.3 Spaces on Simplicial Meshes with DiscontinuousPressure |
123 |
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3.6.4 Spaces on Quadrilateral and Hexahedral Meshes with Discontinuous Pressure |
127 |
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3.6.5 Non-conforming Finite Element Spaces of Lowest Order |
129 |
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3.6.6 Computing the Discrete Inf-Sup Constant |
136 |
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3.7 The Helmholtz Decomposition |
139 |
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4 The Stokes Equations |
148 |
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4.1 The Continuous Equations |
148 |
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4.2 Finite Element Error Analysis |
155 |
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4.2.1 Conforming Inf-Sup Stable Pairs of Finite Element Spaces |
156 |
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4.2.1.1 The Case Vdivh Vdiv |
157 |
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4.2.1.2 The Case Vdivh Vdiv |
171 |
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4.2.2 The Stokes Projection |
174 |
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4.2.3 Lowest Order Non-conforming Inf-Sup Stable Pairs of Finite Element Spaces |
176 |
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4.3 Implementation of Finite Element Methods |
191 |
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4.4 Residual-Based A Posteriori Error Analysis |
198 |
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4.5 Stabilized Finite Element Methods Circumventing the Discrete Inf-Sup Condition |
209 |
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4.5.1 The Pressure Stabilization Petrov–Galerkin (PSPG) Method |
210 |
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4.5.2 Some Other Stabilized Methods |
224 |
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4.6 Improving the Conservation of Mass, Divergence-Free Finite Element Solutions |
228 |
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4.6.1 The Grad-Div Stabilization |
229 |
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4.6.2 Choosing Appropriate Test Functions |
240 |
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4.6.3 Constructing Divergence-Free and Inf-Sup Stable Pairs of Finite Element Spaces |
248 |
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5 The Oseen Equations |
254 |
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5.1 The Continuous Equations |
254 |
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5.2 The Galerkin Finite Element Method |
260 |
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5.3 Residual-Based Stabilizations |
269 |
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5.3.1 The Basic Idea |
269 |
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5.3.2 The SUPG/PSPG/grad-div Stabilization |
272 |
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5.3.3 Other Residual-Based Stabilizations |
298 |
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5.4 Other Stabilized Finite Element Methods |
300 |
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6 The Steady-State Navier–Stokes Equations |
312 |
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6.1 The Continuous Equations |
312 |
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6.1.1 The Strong Form and the Variational Form |
312 |
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6.1.2 The Nonlinear Term |
313 |
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6.1.3 Existence, Uniqueness, and Stability of a Solution |
323 |
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6.2 The Galerkin Finite Element Method |
327 |
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6.3 Iteration Schemes for Solving the Nonlinear Problem |
344 |
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6.4 A Posteriori Error Estimation with the Dual Weighted Residual (DWR) Method |
353 |
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7 The Time-Dependent Navier–Stokes Equations: Laminar Flows |
365 |
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7.1 The Continuous Equations |
365 |
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7.2 Finite Element Error Analysis: The Time-Continuous Case |
387 |
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7.3 Temporal Discretizations Leading to Coupled Problems |
403 |
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7.3.1 ?-Schemes as Discretization in Time |
403 |
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7.3.2 Other Schemes |
419 |
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7.4 Finite Element Error Analysis: The Fully Discrete Case |
420 |
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7.5 Approaches Decoupling Velocity and Pressure: Projection Methods |
441 |
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8 The Time-Dependent Navier–Stokes Equations: Turbulent Flows |
456 |
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8.1 Some Physical and Mathematical Characteristics of Turbulent Incompressible Flows |
457 |
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8.2 Large Eddy Simulation: The Concept of Space Averaging |
467 |
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8.2.1 The Basic Concept of LES, Space Averaging, Convolution with Filters |
467 |
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8.2.2 The Space-Averaged Navier–Stokes Equations in the Case ?=Rd |
472 |
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8.2.3 The Space-Averaged Navier–Stokes Equations in a Bounded Domain |
475 |
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8.2.4 Analysis of the Commutation Error for the Gaussian Filter |
479 |
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8.2.5 Analysis of the Commutation Error for the Box Filter |
486 |
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8.2.6 Summary of the Results Concerning Commutation Errors |
490 |
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8.3 Large Eddy Simulation: The Smagorinsky Model |
491 |
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8.3.1 The Model of the SGS Stress Tensor: Eddy Viscosity Models |
491 |
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8.3.2 Existence and Uniqueness of a Solution of the Continuous Smagorinsky Model |
495 |
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8.3.3 Finite Element Error Analysis for the Time-Continuous Case |
517 |
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8.3.3.1 The Continuous Problem |
518 |
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8.3.3.2 The Finite Element Problem |
525 |
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8.3.4 Variants for Reducing Some Drawbacks of the Smagorinsky Model |
545 |
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8.4 Large Eddy Simulation: Models Based on Approximations in Wave Number Space |
550 |
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8.4.1 Modeling of the Large Scale and Cross Terms |
551 |
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8.4.2 Models for the Subgrid Scale Term |
558 |
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8.4.3 The Resulting Models |
560 |
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8.5 Large Eddy Simulation: Approximate Deconvolution Models (ADMs) |
562 |
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8.6 The Leray-? Model |
571 |
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8.6.1 The Continuous Problem |
572 |
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8.6.2 The Discrete Problem |
575 |
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8.7 The Navier–Stokes-? Model |
584 |
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8.8 Variational Multiscale Methods |
599 |
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8.8.1 Basic Concepts |
600 |
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8.8.1.1 Two-Scale VMS Methods |
600 |
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8.8.1.2 Three-Scale VMS Methods |
602 |
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8.8.2 A Two-Scale Residual-Based VMS Method |
604 |
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8.8.3 A Two-Scale VMS Method with Time-Dependent Orthogonal Subscales |
612 |
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8.8.4 A Three-Scale Bubble VMS Method |
619 |
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8.8.5 Three-Scale Algebraic Variational Multiscale-Multigrid Methods (AVM3 and AVM4) |
623 |
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8.8.6 A Three-Scale Coarse Space Projection-Based VMS Method |
628 |
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8.8.6.1 Definition of the Method |
628 |
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8.8.6.2 Imbedding the Method into the Basic Approach From Sect.8.8.1 |
630 |
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8.8.6.3 Finite Element Error Analysis |
632 |
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8.8.6.4 Implementation and Numerical Experience |
643 |
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8.9 Comparison of Some Turbulence Models in Numerical Studies |
649 |
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9 Solvers for the Coupled Linear Systems of Equations |
657 |
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9.1 Solvers for the Coupled Problems |
658 |
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9.2 Preconditioners for Iterative Solvers |
660 |
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9.2.1 Incomplete Factorizations |
661 |
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9.2.2 A Coupled Multigrid Method |
662 |
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9.2.3 Preconditioners Treating Velocity and Pressure in a Decoupled Way |
674 |
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A Functional Analysis |
684 |
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A.1 Metric Spaces, Banach Spaces, and Hilbert Spaces |
684 |
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A.2 Function Spaces |
688 |
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A.3 Some Definitions, Statements, and Theorems |
696 |
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B Finite Element Methods |
706 |
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B.1 The Ritz Method and the Galerkin Method |
706 |
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B.2 Finite Element Spaces |
714 |
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B.3 Finite Elements on Simplices |
718 |
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B.4 Finite Elements on Parallelepipeds and Quadrilaterals |
726 |
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B.5 Transform of Integrals |
732 |
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C Interpolation |
735 |
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C.1 Interpolation in Sobolev Spaces by Polynomials |
735 |
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C.2 Interpolation of Non-smooth Functions |
745 |
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C.3 Orthogonal Projections |
749 |
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C.4 Inverse Estimate |
751 |
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D Examples for Numerical Simulations |
754 |
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D.1 Examples for Steady-State Flow Problems |
757 |
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D.2 Examples for Laminar Time-Dependent Flow Problems |
765 |
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D.3 Examples for Turbulent Flow Problems |
772 |
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E Notations |
782 |
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References |
790 |
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Index of Subjects |
809 |
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