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1 Introduction1

1.1 Types of PDEs1

1.2 Grids and Discretization Approaches3

1.2.1 Grids3

1.2.2 Discretization Approaches6

1.3 SomeNotation7

1.3.1 Continuous Boundary Value Problems8

1.3.2 Discrete Boundary Value Problems8

1.3.3 Inner Products and Norms9

1.3.4 Stencil Notation10

1.4 Poisson's Equation and Model Problem 110

1.4.1 Matrix Terminology12

1.4.2 Poisson Solvers14

1.5 A First Glance at Multigrid15

1.5.1 The Two Ingredients of Multigrid15

1.5.2 High and Low Frequencies,and Coarse Meshes17

1.5.3 From Two Grids to Multigrid19

1.5.4 Multigrid Features20

1.5.5 Multigrid History23

1.6 Intermezzo:Some Basic Facts and Methods24

1.6.1 Iterative Solvers,Splittings and Preconditioners24

2 Basic Multigrid I28

2.1 Error Smoothing Procedures28

2.1.1 Jacobi-type Iteration(Relaxation)29

2.1.2 Smoothing Properties of ω-Jacobi Relaxation30

2.1.3 Gauss-Seidel-type Iteration(Relaxation)31

2.1.4 Parallel Properties of Smoothers33

2.2 Introducing the Two-grid Cycle34

2.2.1 Iteration by Approximate Solution of the Defect Equation35

2.2.2 Coarse Grid Correction37

2.2.3 Structure of the Two-grid Operator39

2.3 Multigrid Components41

2.3.1 Choices of Coarse Grids41

2.3.2 Choice of the Coarse Grid Operator42

2.3.3 Transfer Operators:Restriction42

2.3.4 Transfer Operators:Interpolation43

2.4 The Multigrid Cycle45

2.4.1 Sequences of Grids and Operators46

2.4.2 Recursive Definition46

2.4.3 Computational Work50

2.5 Multigrid Convergence and Efficiency52

2.5.1 An Efficient 2D Multigrid Poisson Solver52

2.5.2 How to Measure the Multigrid Convergence Factor in Practice54

2.5.3 Numerical Efficiency55

2.6 Full Multigrid56

2.6.1 Structure of Full Multigrid57

2.6.2 Computational Work59

2.6.3 FMG for Poisson's Equation59

2.7 Further Remarks on Transfer Operators60

2.8 First Generalizations62

2.8.1 2D Poisson-like Differential Equations62

2.8.2 Time-dependent Problems63

2.8.3 Cartesian Grids in Nonrectangular Domains66

2.8.4 Multigrid Components for Cell-centered Discretizations69

2.9 Multigrid in 3D70

2.9.1 The 3D Poisson Problem70

2.9.2 3D Multigrid Components71

2.9.3 Computational Work in 3D74

3 Elementary Multigrid Theory75

3.1 Survey76

3.2 Why it is Sufficient to Derive Two-grid Convergence Factors77

3.2.1 h-Independent Convergence of Multigrid77

3.2.2 A Theoretical Estimate for Full Multigrid79

3.3 How to Derive Two-grid Convergence Factors by Rigorous Fourier Analysis82

3.3.1 Asymptotic Two-grid Convergence82

3.3.2 Norms of the Two-grid Operator83

3.3.3 Results for Multigrid85

3.3.4 Essential Steps and Details of the Two-grid Analysis85

3.4 Range of Applicability of the Rigorous Fourier Analysis,Other Approaches91

3.4.1 The 3D Case91

3.4.2 Boundary Conditions93

3.4.3 List of Applications and Limitations93

3.4.4 Towards Local Fourier Analysis94

3.4.5 Smoothing and Approximation Property:a Theoretical Overview96

4 Local Fourier Analysis98

4.1 Background99

4.2 Terminology100

4.3 Smoothing Analysis Ⅰ102

4.4 Two-grid Analysis106

4.5 Smoothing Analysis Ⅱ113

4.5.1 Local Fourier Analysis for GS-RB115

4.6 Some Results,Remarks and Extensions116

4.6.1 Some Local Fourier Analysis Results for Model Problem 1117

4.6.2 Additional Remarks118

4.6.3 Other Coarsening Strategies121

4.7 h-Ellipticity121

4.7.1 The Concept of h-Ellipticity123

4.7.2 Smoothing and h-Ellipticity126

5 Basic Multigrid Ⅱ130

5.1 Anisotropic Equations in 2D131

5.1.1 Failure of Pointwise Relaxation and Standard Coarsening131

5.1.2 Semicoarsening133

5.1.3 Line Smoothers134

5.1.4 Strong Coupling of Unknowns in Two Directions137

5.1.5 An Example with Varying Coefficients139

5.2 Anisotropic Equations in 3D141

5.2.1 Standard Coarsening for 3D Anisotropic Problems143

5.2.2 Point Relaxation for 3D Anisotropic Problems145

5.2.3 Further Approaches,Robust Variants147

5.3 Nonlinear Problems,the Full Approximation Scheme147

5.3.1 Classical Numerical Methods for Nonlinear PDEs:an Example148

5.3.2 Local Linearization151

5.3.3 Linear Multigrid in Connection with Global Linearization153

5.3.4 Nonlinear Multigrid:the Full Approximation Scheme155

5.3.5 Smoothing Analysis:a Simple Example159

5.3.6 FAS for the Full Potential Equation160

5.3.7 The(h,H)-Relative Truncation Error and τ-Extrapolation163

5.4 Higher Order Discretizations166

5.4.1 Defect Correction168

5.4.2 The Mehrstellen Discretization for Poisson's Equation172

5.5 Domains with Geometric Singularities174

5.6 Boundary Conditions and Singular Systems177

5.6.1 General Treatment of Boundary Conditions in Multigrid178

5.6.2 Neumann Boundary Conditions179

5.6.3 Periodic Boundary Conditions and Global Constraints183

5.6.4 General Treatment of Singular Systems185

5.7 Finite Volume Discretization and Curvilinear Grids187

5.8 General Grid Structures190

6 Parallel Multigrid in Practice193

6.1 Parallelism of Multigrid Components194

6.1.1 Parallel Components for Poisson's Equation195

6.1.2 Parallel Complexity196

6.2 Grid Partitioning197

6.2.1 Parallel Systems,Processes and Basic Rules for Parallelization198

6.2.2 Grid Partitioning for Jacobi and Red-Black Relaxation199

6.2.3 Speed-up and Parallel Efficiency204

6.2.4 A Simple Communication Model206

6.2.5 Scalability and the Boundary-volume Effect207

6.3 Grid Partitioning and Multigrid208

6.3.1 Two-grid and Basic Multigrid Considerations208

6.3.2 Multigrid and the Verv Coarse Grids211

6.3.3 Boundary-volume Effect and Scalability in the Multigrid Context214

6.3.4 Programming Parallel Systems215

6.4 Parallel Line Smoothers216

6.4.1 1D Reduction(or Cyclic Reduction)Methods217

6.4.2 Cyclic Reduction and Grid Partitioning218

6.4.3 Parallel Plane Relaxation220

6.5 Modifications of Multigrid and Related Approaches221

6.5.1 Domain Decomposition Methods:a Brief Survey221

6.5.2 Multigrid Related Parallel Approaches225

7 More Advanced Multigrid227

7.1 The Convection-Diffusion Equation:Discretization Ⅰ228

7.1.1 The 1D Case228

7.1.2 Central Differencing230

7.1.3 First-order Upwind Discretizations and Artificial Viscosity233

7.2 The Convection-Diffusion Equation:Multigrid Ⅰ234

7.2.1 Smoothers for First-order Upwind Discretizations235

7.2.2 Variable Coefficients237

7.2.3 The Coarse Grid Correction239

7.3 The Convection-Diffusion Equation:Discretization Ⅱ243

7.3.1 Combining Central and Upwind Differencing243

7.3.2 Higher Order Upwind Discretizations244

7.4 The Convection-Diffusion Equation:Multigrid Ⅱ249

7.4.1 Line Smoothers for Higher Order Upwind Discretizations249

7.4.2 Multistage Smoothers253

7.5 ILU Smoothing Methods256

7.5.1 Idea of ILU Smoothing257

7.5.2 Stencil Noration259

7.5.3 ILU Smoothing for the Anisotropic Diffusion Equation261

7.5.4 A Particularly Robust ILU Smoother262

7.6 Problems with Mixed Derivatives263

7.6.1 Standard Smoothing and Coarse Grid Correction264

7.6.2 ILU Smoothing267

7.7 Problems with Jumping Coefficients and Galerkin Coarse Grid Operators268

7.7.1 Jumping Coefficients269

7.7.2 Multigrid for Problems with Jumping Coefficients271

7.7.3 Operator-dependent Interpolation272

7.7.4 The Galerkin Coarse Grid Operator273

7.7.5 Further Remarks on Galerkin-based Coarsening277

7.8 Multigrid as a Preconditioner(Acceleration of Multigrid by Iterant Recombination)278

7.8.1 The Recirculating Convection-Diffusion Problem Revisited278

7.8.2 Multigrid Acceleration by Iterant Recombination280

7.8.3 Krylov Subspace Iteration and Multigrid Preconditioning282

7.8.4 Multigrid:Solver versus Preconditioner287

8 Multigrid for Systems of Equations289

8.1 Notation and Introductory Remarks290

8.2 Multigrid Components293

8.2.1 Restriction293

8.2.2 Interpolation of Coarse Grid Corrections294

8.2.3 Orders of Restriction and Interpolation295

8.2.4 Solution on the Coarsest Grid295

8.2.5 Smoothers295

8.2.6 Treatment of Boundary Conditions296

8.3 LFA for Systems of PDEs297

8.3.1 Smoothing Analysis297

8.3.2 Smoothing and h-Ellipticity300

8.4 The Biharmonic System301

8.4.1 A Simple Example:GS-LEX Smoothing302

8.4.2 Treatment of Boundary Conditions303

8.4.3 Multigrid Convergence304

8.5 A Linear Shell Problem307

8.5.1 Decoupled Smoothing308

8.5.2 Collective versus Decoupled Smoothing310

8.5.3 Level-dependent Smoothing311

8.6 Introduction to Incompressible Navier-Stokes Equations312

8.6.1 Equations and Boundary Conditions312

8.6.2 Survey314

8.6.3 The Checkerboard Instability315

8.7 Incompressible Navier-Stokes Equations:Staggered Discretizations316

8.7.1 Transfer Operators318

8.7.2 Box Smoothing320

8.7.3 Distributive Smoothing323

8.8 Incompressible Navier-Stokes Equations:Nonstaggered Discretizations326

8.8.1 Artificial Pressure Terms327

8.8.2 Box Smoothing328

8.8.3 Alternative Formulations331

8.8.4 Flux Splitting Concepts333

8.8.5 Flux Difference Splitting and Multigrid:Examples338

8.9 Compressible Euler Equations343

8.9.1 Introduction345

8.9.2 Finite Volume Discretization and Appropriate Smoothers347

8.9.3 Some Examples349

8.9.4 Multistage Smoothers in CFD Applications352

8.9.5 Towards Compressible Navier-Stokes Equations354

9 Adaptive Multigrid356

9.1 A Simple Example and Some Notation357

9.1.1 A Simple Example357

9.1.2 Hierarchy of Grids360

9.2 The Idea of Adaptive Multigrid361

9.2.1 The Two-grid Case361

9.2.2 From Two Grids to Multigrid363

9.2.3 Self-adaptive Full Multigrid364

9.3 Adaptive Multigrid and the Composite Grid366

9.3.1 Conservative Discretization at Interfaces of Refinement Areas367

9.3.2 Conservative Interpolation369

9.3.3 The Adaptive Multigrid Cycle372

9.3.4 Further Approaches373

9.4 Refinement Criteria and Optimal Grids373

9.4.1 Refinement Criteria374

9.4.2 Optimal Grids and Computational Effort378

9.5 Parallel Adaptive Multigrid379

9.5.1 Parallelization Aspects379

9.5.2 Distribution of Locally Refined Grids381

9.6 Some Practical Results382

9.6.1 2D Euler Flow Around an Airfoil382

9.6.2 2D and 3D Incompressible Navier-Stokes Equations385

10 Some More Multigrid Applications389

10.1 Multigrid for Poisson-type Equations on the Surface of the Sphere389

10.1.1 Discretization390

10.1.2 Specific Multigrid Components on the Surface of a Sphere391

10.1.3 A Segment Relaxation393

10.2 Multigrid and Continuation Methods395

10.2.1 The Bratu Problem396

10.2.2 Continuation Techniques397

10.2.3 Multigrid for Continuation Methods398

10.2.4 The Indefinite Helmholtz Equation400

10.3 Generation of Boundary Fitted Grids400

10.3.1 Grid Generation Based on Poisson's Equation401

10.3.2 Multigrid Solution of Grid Generation Equations401

10.3.3 Grid Generation with the Biharmonic Equation403

10.3.4 Examples404

10.4 LiSS:a Generic Multigrid Software Package404

10.4.1 Pre-and Postprocessing Components405

10.4.2 The Multigrid Solver406

10.5 Multigrid in the Aerodynamic Industry407

10.5.1 Numerical Challenges for the 3D Compressible Navier-Stokes Equations408

10.5.2 FLOWer409

10.5.3 Fluid-Structure Coupling410

10.6 How to Continue with Multigrid411

Appendixes413

A An Introduction to Algebraic Multigrid(by Klaus Stüben)413

A.1 Introduction413

A.1.1 Geometric Multigrid414

A.1.2 Algebraic Multigrid415

A.1.3 An Example418

A.1.4 Overview of the Appendix420

A.2 Theoretical Basis and Notation422

A.2.1 Formal Algebraic Multigrid Components422

A.2.2 Further Notation425

A.2.3 Limit Case of Direct Solvers427

A.2.4 The Variational Principle for Positive Definite Problems430

A.3 Algebraic Smoothing432

A.3.1 Basic Norms and Smooth Eigenvectors433

A.3.2 Smoothing Property of Relaxation434

A.3.3 Interpretation of Algebraically Smooth Error438

A.4 Postsmoothing and Two-level Convergence444

A.4.1 Convergence Estimate445

A.4.2 Direct Interpolation447

A.4.3 Indirect Interpolation459

A.5 Presmoothing and Two-level Convergence461

A.5.1 Convergence using Mere F-Smoothing461

A.5.2 Convergence using Full Smoothing468

A.6 Limits of the Theory469

A.7 The Algebraic Multigrid Algorithm472

A.7.1 Coarsening473

A.7.2 Interpolation479

A.7.3 Algebraic Multigrid as Preconditioner484

A.8 Applications485

A.8.1 Default Settings and Notation486

A.8.2 Poisson-like Problems487

A.8.3 Computational Fluid Dynamics494

A.8.4 Problems with Discontinuous Coefficients503

A.8.5 Further Model Problems513

A.9 Aggregation-based Algebraic Multigrid522

A.9.1 Rescaling of the Galerkin Operator524

A.9.2 Smoothed Aggregation526

A.10 Further Developments and Conclusions528

B Subspace Correction Methods and Multigrid Theory(by Peter Oswald)533

B.1 Introduction533

B.2 Space Splittings536

B.3 Convergence Theory551

B.4 Multigrid Applications556

B.5 A Domain Decomposition Example561

C Recent Developments in Multigrid Efficiency in Computational Fluid Dynamics(by Achi Brandt)573

C.1 Introduction573

C.2 Table of Difficulties and Possible Solutions574

References590

Index613