多重网格

Preface1 IⅡtrOduction 1.1 Types ofPDEs 1.2 Grids and Discretization Approaches 1.2.1 Grids 1.2.2 Discretization Approaches 1.3 Some Notation 1.3.1 Continuous Boundary Value Problems l.3.2 Discrete Boundary Value Problems 1.3.3 Inner Products and NOrlTIS 1.3.4 Stencil Notation 1.4 Poisson's Equation and Model Problem 1 1.4.1 Matrix Terminology 1.4.2 Poisson Solvers 1.5 A First Glance at Multigrid 1.5.1 The Two Ingredients ofMultigrid 1.5.2 High and Low Frequencies，and Coarse Meshes 1.5.3 From Two Grids to Multigrid 1.5.4 Multigrid Features 1.5.5 Multigrid History 1.6 Intermezzo：Some Basic Facts and Methods 1.6.1 Iterative Solvers.Splittings and Preconditioners2 Basic Multigrid I 2.1 Error Smoothing Procedures 2.1.1 Jacobi.type Iteration(Relaxation) 2.1.2 Smoothing Properties of山一Jacobi Relaxation 2.1.3 Gauss.Seidel.type Iteration(Relaxation) 2.1.4 Parallel Properties of Smoothers 2.2 Introducing the Two.grid Cycle 2.2.1 Iteration by Approximate Solution ofthe Defect Equation 2.2.2 Coarse Grid Correction 2.2.3 Structure ofthe Two—grid Operator 2.3 Multigrid Components 2.3.1 Choices of Coarse Grids 2.3.2 Choice of the Coarse Grid Operator 2.3.3 Transfer Operators：Restriction 2.3.4 Transfer Operators：Interpolation 2.4 The Multigrid Cycle 2.4.1 Sequences of Grids and Operators 2.4.2 Recursive Definition 2.4.3 Computational W.0rk 2.5 Multigrid Convergence and E币ciency 2.5.1 An Efficient 2D Multigrid Poisson Solver 2.5.2 How to Measure the Multigrid Convergence Factor in Practice 2.5.3 Numerical E伍ciency 2.6 Full Multigrid 2.6.1 Structure of Full Multigrid 2.6.2 Computational W10rk 2.6.3 FMG for Poisson'S Equation 2.7 Further Remarks on Transfer Operators 2.8 First Generalizations 2.8.1 2D Poisson—like Difrerential Equations 2.8.2 Time.dependent Problems 2.8.3 Cartesian Grids in Nonrectangular Domains 2.8.4 Multigrid Components for Cell.centered Discretizations 2.9 Multigridin 3D 2.9.1 TIle 3D Poisson Problem 2.9.2 3D Multigrid Components 2.9.3 Computational WOrk in 3D 3 Elementary Multigrid Theory 3.1 Survey 3.2 Why it iS Su伍cient tO Derive Two.grid Convergence Factors 3.2.1 Independent Convergence of Multigrid 3.2.2 A 111eoretical Estimate for Full Multigrid 3.3 How to Derive Two—grid Convergence Factors by Rigorous Fourier Analysis 3.3.1 Asymptotic Two.grid Convergence. 3.3.2 Norms ofthe Two—grid Operator 3.3.3 Results for Multigrid 3.3.4 Essential Steps and Details of the Two.grid Analysis 3.4 Range of Applicability of the Rigorous Fourier Analysis，Other Approaches 3.4.1 The 3D Case …… 4 Local Fourier Analysis5 Basic Multigrid II6 ParalleI Multigrid in Practice7 More Advanced Multigrid8 Multigrid for Systems of Equations9 Adaptive Muitigrid10 Some More Multigrid ApplicationsADDendixes