Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- List of Figures -- Preface -- 1 Fluid Mechanics and Computation: An Introduction -- 1.1 Viscous Fluid Flows -- 1.2 Mass Conservation -- 1.3 Momentum Equations -- 1.3.1 Linear Momentum -- 1.3.2 Angular Momentum -- 1.4 Energy Conservation -- 1.5 Thermodynamics and Constitutive Equations -- 1.6 Fluid Flow Equations and Boundary Conditions -- 1.6.1 Isothermal Incompressible Flow -- 1.6.2 Thermal Convection: The Boussinesq Approximation -- 1.6.3 Boundary and Initial Conditions -- 1.7 Dimensional Analysis and Reduced Equations -- 1.8 Vorticity Equation -- 1.9 Simplified Models -- 1.10 Turbulence and Challenges -- 1.11 Numerical Simulation -- 1.11.1 Hardware Issues -- 1.11.2 Software Issues -- 1.11.3 Algorithms -- 1.11.4 Advantages of High-Order Methods -- 2 Approximation Methods for Elliptic Problems -- 2.1 Variational Form of Boundary-Value Problems -- 2.1.1 Variational Functionals -- 2.1.2 Boundary Conditions -- 2.1.3 Sobolev Spaces and the Lax-Milgram Theorem -- Lemma 2.1 (Poincaré-Friedrichs inequality) -- 2.2 An Approximation Framework -- 2.2.1 Galerkin Approximations -- Galerkin Method -- Petrov-Galerkin Methods -- Generalized Galerkin Method -- 2.2.2 Collocation Approximation -- 2.3 Finite-Element Methods -- 2.3.1 The h-Version of Finite Elements -- Global Mesh and Set of Basis Functions -- Stiffness and Mass Matrices -- 2.3.2 The p-Version of Finite Elements -- 2.4 Spectral-Element Methods -- 2.5 Orthogonal Collocation -- 2.5.1 Orthogonal Collocation in a Monodomain -- 2.5.2 Orthogonal Collocation in a Multidomain -- 2.6 Error Estimation -- 2.7 Solution Techniques -- 2.7.1 The Conditioning of a Matrix -- Matrix Norms and Spectral Radius -- The Condition Number -- The Conditioning of Spectral Schemes -- 2.7.2 Basic Iterative Methods -- Convergence Results.

2.7.3 Preconditioning Schemes of High-Order Methods -- 2.7.4 Iterative Methods Based on Projection -- 2.8 A Numerical Example -- 3 Parabolic and Hyperbolic Problems -- 3.1 Introduction -- 3.2 Time Discretization Schemes -- 3.2.1 Linear Multistep Methods -- Main Theoretical Concepts -- Adams-Bashforth and Adams-Moulton Schemes -- Backward Differencing Schemes -- 3.2.2 Predictor-Corrector Methods -- 3.2.3 Runge-Kutta Methods -- Explicit Runge-Kutta Schemes -- Implicit Runge-Kutta Schemes -- 3.3 Splitting Methods -- 3.3.1 The Operator-Integration-Factor Splitting Method -- 3.3.2 OIFS Example: The BDF3/RK4 Scheme -- 3.4 The Parabolic Case: Unsteady Diffusion -- 3.4.1 Spatial Discretization -- 3.4.2 Time Advancement -- 3.5 The Hyperbolic Case: Linear Convection -- 3.5.1 Spatial Discretization -- 3.5.2 Eigenvalues of the Discrete Problem and CFL Number -- 3.5.3 Example of Temporal and Spatial Accuracy -- 3.5.4 Inflow-Outflow Boundary Conditions -- 3.6 Steady Advection-Diffusion Problems -- 3.6.1 Spectral Elements and Bubble Stabilization -- 3.6.2 Collocation and Staggered Grids -- 3.7 Unsteady Advection-Diffusion Problems -- 3.7.1 Spatial Discretization -- 3.7.2 Temporal Discretization -- 3.7.3 Outflow Conditions and Filter-Based Stabilization -- 3.8 The Burgers Equation -- 3.8.1 Space and Time Discretization -- Orthogonal Collocation -- Spectral-Element Method -- 3.8.2 Numerical Results -- 3.9 The OIFS Method and Subcycling -- 3.10 Taylor-Galerkin Time Integration -- 3.10.1 Nonlinear Pure Advection -- 3.10.2 Taylor-Galerkin and OIFS Methods -- 4 Multidimensional Problems -- 4.1 Introduction -- 4.2 Tensor Products -- Operator Evaluation -- 4.3 Elliptic Problems -- 4.3.1 Weak Formulation and Sobolev Spaces -- 4.3.2 A Constant-Coefficient Case -- 4.3.3 The Variable-Coefficient Case -- 4.4 Deformed Geometries -- 4.4.1 Generation of Geometric Deformation.

4.4.2 Surface Integrals and Robin Boundary Conditions -- 4.5 Spectral-Element Discretizations -- 4.5.1 Continuity and Direct Stiffness Summation -- 4.5.2 Spectral-Element Operators -- 4.5.3 Inhomogeneous Dirichlet Problems -- 4.5.4 Iterative Solution Techniques -- 4.5.5 Two-Dimensional Examples -- 4.6 Collocation Discretizations -- 4.6.1 The Diffusion Case -- Inhomogeneous Neumann Boundary Conditions -- The Multidomain Approach -- 4.6.2 The Advection-Diffusion Case -- 4.7 Parabolic Problems -- 4.7.1 Time-Dependent Projection -- 4.7.2 Other Diffusion Systems -- 4.8 Hyperbolic Problems -- A Two-Dimensional Example -- 4.9 Unsteady Advection-Diffusion Problems -- 4.10 Further Reading -- 5 Steady Stokes and Navier-Stokes Equations -- 5.1 Steady Velocity-Pressure Formulation -- 5.2 Stokes Equations -- 5.2.1 The Weak Formulation -- 5.2.2 The Spectral-Element Method -- Staggered Spectral Elements -- Collocative Spectral Elements -- 5.2.3 Collocation Methods on Single and Staggered Grids -- Legendre Single-Grid Collocation -- Chebyshev Single-Grid Collocation -- Legendre Staggered-Grid Collocation -- 5.3 Linear Systems, Algorithms, and Preconditioners -- 5.3.1 Spectral-Element Methods and Uzawa Algorithm -- 5.3.2 Collocation Methods -- 5.4 Poisson Pressure Solver and Green's-Function Technique -- 5.4.1 General Considerations -- 5.4.2 The Green's-Function Method -- 5.4.3 Implementation -- 5.5 Divergence-Free Bases -- 5.6 Stabilization of the PN-PN Approximation by Bubble Functions -- 5.7 hp-Methods for Stokes Problems -- 5.8 Steady Navier-Stokes Equations -- 5.8.1 Weak Formulation -- 5.8.2 Collocation Approximation of the Navier-Stokes Equations -- 5.8.3 Solution Algorithms: Iterative and Newton Methods -- 5.9 Applications -- 5.9.1 Stokes Problems -- Square-Cavity Problem -- Flow in a Wedge -- Grooved Channel -- Stabilization by Bubble Functions.

Wannier-Stokes Flow -- 5.9.2 Navier-Stokes Problems -- Kovasznay Flow -- Grooved Channel -- Cooled Hot Cylinder -- 5.10 Complements and Engineering Considerations -- 6 Unsteady Stokes and Navier-Stokes Equations -- 6.1 Unsteady Velocity-Pressure Formulation -- 6.2 Unsteady Stokes Equations -- 6.2.1 The Weak Formulation -- 6.2.2 Uzawa Algorithm -- 6.2.3 Splitting and Decoupling Algorithms -- 6.3 Pressure Preconditioning -- 6.4 Unsteady Navier-Stokes Equations -- 6.4.1 Weak Formulation -- 6.4.2 Advection Treatment -- 6.5 Projection Methods -- 6.5.1 Fractional-Step Method -- 6.5.2 Pressure Correction Method -- 6.6 Stabilizing Unsteady Flows -- 6.7 Arbitrary Lagrangian-Eulerian Formulation and Free-Surface Flows -- 6.7.1 ALE Formulation -- 6.7.2 Free-Surface Conditions -- 6.7.3 Variational Formulation of Free-Surface Flows -- 6.7.4 Space and Time Discretization -- 6.8 Unsteady Applications -- 6.8.1 Extrusion from a Die -- 6.8.2 Vortex-Sheet Roll-Up -- 6.8.3 Unsteady Flow in Arteriovenous Grafts -- 6.9 Further Reading and Engineering Considerations -- 7 Domain Decomposition -- 7.1 Introduction -- 7.2 Preconditioning Methods -- 7.2.1 Substructuring and the Steklov-Poincaré Operator -- The Continuous Presentation -- The Discrete Presentation -- 7.2.2 Overlapping Schwarz Procedures -- Discrete Formulation -- Schwarz Preconditioners -- A Brief Analysis -- Two-Level Preconditioners -- 7.2.3 Schwarz Preconditioners for High-Order Methods -- 7.2.4 Spectral-Element Multigrid -- 7.3 The Mortar Element Method -- 7.3.1 Elliptic Problems -- 7.3.2 Implementation -- 7.3.3 Steady Stokes Problems -- 7.3.4 Applications -- Flow Around an Impeller -- Resonator Cavity Flow -- Clearance-Gap Glow -- 7.4 Adaptivity and Singularity Treatment -- 7.4.1 Coupling between Finite and Spectral Elements -- 7.4.2 Singularity Treatment -- 7.4.3 Triangular and Tetrahedral Elements.

Modal Bases -- Integration -- Differentiation -- Nodal Bases -- 7.4.4 Error Estimates and Adaptivity -- Spectral Error Estimator -- Physical Error Indicators -- 7.5 Further Reading -- 8 Vector and Parallel Implementations -- 8.1 Introduction -- 8.2 Serial Architectures -- 8.2.1 Pipelining -- 8.2.2 Memory, Bandwidth, and Caches -- 8.3 Tensor-Product Operator Evaluation -- 8.3.1 Tensor-Product Evaluation -- 8.3.2 Other Operations -- 8.4 Parallel Programming -- 8.4.1 Communication Characteristics -- 8.4.2 Vector Reductions -- 8.5 Parallel Multidomain Methods -- 8.5.1 Data Distribution and Operator Evaluation -- 8.5.2 Direct Stiffness Summation -- 8.5.3 Domain Partitioning -- 8.5.4 Coarse-Grid Solves -- 8.6 Applications -- 8.6.1 Hairpin Vortices -- 8.6.2 Driven Cavity -- 8.6.3 Backward-Facing Step -- 8.7 Further Reading -- Appendix A Preliminary Mathematical Concepts -- A.1 Metric Spaces -- A.1.1 Definition -- Examples of Metric Spaces -- A.1.2 Open Set, Closed Set, Neighborhood -- A.1.3 Cauchy Sequence, Limit Points, Dense Sets -- A.1.4 Mapping, Domain, Range, Continuity -- A.1.5 Convergence, Completeness, Completion Process -- A.2 Normed Spaces -- A.2.1 Definition -- Examples of Normed Spaces -- A.2.2 Banach Spaces -- A.3 Linear Operators and Functionals in Normed Spaces -- A.3.1 Linear Operator, Domain, Range, Nullspace -- A.3.2 The Inverse Operator -- Example: Matrices -- A.3.3 Bounded Operators, Compact Operators -- A.3.4 Bounded Linear Functionals, Dual Spaces -- A.3.5 The Fréchet Derivative of an Operator -- Examples of Fréchet Derivatives -- A.4 Inner-Product Spaces -- A.4.1 Definition -- Examples of Inner-Product Spaces -- A.4.2 Hilbert Spaces -- Examples of Hilbert Spaces -- A.4.3 Cauchy-Schwarz Inequality -- A.4.4 The Riesz Representation -- A.4.5 Orthogonality, Orthogonal Projection -- A.4.6 Separable Hilbert Spaces, Basis.

A.4.7 Gram-Schmidt Orthonormalization Process.