Finite Element Methods for Incompressible Flow Problems (eBook)

Volker John (Autor)

eBook Download: PDF

2016 | 1st ed. 2016
XIII, 812 Seiten
Springer International Publishing (Verlag)
978-3-319-45750-5 (ISBN)

Infos

This book explores finite element methods for incompressible flow problems: Stokes equations, stationary Navier-Stokes equations and time-dependent Navier-Stokes equations. It focuses on numerical analysis, but also discusses the practical use of these methods and includes numerical illustrations. It also provides a comprehensive overview of analytical results for turbulence models. The proofs are presented step by step, allowing readers to more easily understand the analytical techniques.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 27.10.2016
• Reihe/Serie: Springer Series in Computational Mathematics
• Reihe/Serie: Springer Series in Computational Mathematics
• Zusatzinfo: XIII, 812 p. 120 illus., 85 illus. in color.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: finite element analysis • Partial differential equations • saddle point problems • Stokes and Navier-Stokes equations • support of analysis by numerical results • turbulence problems
• ISBN-10: 3-319-45750-0 / 3319457500
• ISBN-13: 978-3-319-45750-5 / 9783319457505