Birkhoff / Schoenstadt Elliptic Problem Solvers

Volume II
1. Auflage 2014
ISBN: 978-1-4832-6339-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 588 Seiten, Web PDF

Elliptic Problem Solvers, II covers the proceedings of the Elliptic Problem Solvers Conference, held at the Naval Postgraduate School in Monterey, California from January 10 to 12, 1983. The book focuses on various aspects of the numerical solution of elliptic boundary value problems. The selection first offers information on building elliptic problem solvers with ELLPACK; presentation and evolution of the club module; and a fourth order accurate fast direct method for the Helmholtz equation. The text then examines the ITPACK project, CMMPAK, solving elliptic problems on an array processor system, and parallel architectures for iterative methods on adaptive, block structured grids. Topics include adaptive solution algorithm, data structure, elliptic problem solvers, input data, and vector ITPACK. The publication ponders on conjugate gradient preconditioners for vector and parallel processors; an algebra for systolic computation; and an incomplete-Cholesky factorization by a matrix partition algorithm. The book also tackles the numerical solution of a model equation near the onset of the Rayleigh-Benard instability; numerical methods for solving coupled semiconductor equations on a minicomputer; and analysis of nonlinear elliptic systems arising in reaction/diffusion modeling. The selection is highly recommended for researchers interested in elliptic problem solvers.

Autoren/Hrsg.

Birkhoff, Garrett

Schoenstadt, Arthur

Inhaltsverzeichnis

1;Front Cover;1
2;Elliptic Problem Solvers II;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;10
6;Preface;14
7;Part I: Software Packages;16
7.1;CHAPTER 1. BUILDING ELLIPTIC PROBLEM SOLVERS WITH ELLPACK;18
7.1.1;I. INTRODUCTION;18
7.1.2;II. THE ELLPACK SYSTEM;19
7.1.3;III. NONLINEAR PROBLEMS AND PICARD S METHOD;20
7.1.4;IV. NONLINEAR PROBLEMS AND NEWTON'S METHOD;23
7.1.5;V. INITIAL VALUE PROBLEMS;28
7.1.6;VI. SYSTEMS OF ELLIPTIC PROBLEMS;32
7.1.7;REFERENCES;36
7.2;CHAPTER 2. PRESENTATION AND EVOLUTION OF THE CLUB MODULEF: A LIBRARY OF COMPUTER PROCEDURES FOR FINITE
ELEMENT ANALYSIS;38
7.2.1;I. THE CLUB MODULEF;38
7.2.2;II. SOME MODULEF PRODUCTIONS;41
7.2.3;III. MODULEF: ADVANCES AND DEVELOPMENT;46
7.2.4;REFERENCES;48
7.3;CHAPTER 3. A FOURTH ORDER ACCURATE FAST DIRECT METHOD
FOR THE HELMHOLTZ EQUATION;50
7.3.1;I. INTRODUCTION;50
7.3.2;II. THE FOURIER ALGORITHM;52
7.3.3;III. HIGH ORDER COMPACT DIFFERENCES;53
7.3.4;IV. THE THREE–DIMENSIONAL
CASE;55
7.3.5;V. SOFTWARE AND COMPUTATIONAL EXAMPLES;56
7.3.6;REFERENCES;59
7.4;CHAPTER 4. The (New) Yale Sparse Matrix Package;60
7.4.1;1. Introduction;60
7.4.2;2. The (Old) Yale Sparse Matrix Package;60
7.4.3;3. The (New) Yale Sparse Matrix Package;63
7.4.4;References;66
7.5;CHAPTER 5. THE ITPACK PROJECT: PAST, PRESENT, AND FUTURE;68
7.5.1;I. INTRODUCTION;68
7.5.2;II.
PAST;69
7.5.3;III.
PRESENT;70
7.5.4;IV. ITPACK IN ELLPACK;71
7.5.5;V. FUTURE;73
7.5.6;VI. VECTOR ITPACK;74
7.5.7;VII. ACKNOWLEDGMENT;76
7.5.8;VIII. REFERENCES;76
7.6;CHAPTER 6. CMMPAK - THE CAPACITANCE MATRIX SOFTWARE PACKAGE;80
7.6.1;INITIALIZE;85
7.6.2;INPUT DATA;86
7.6.3;CALL CMMIMP;86
7.6.4;PRINTOUT;86
7.6.5;REFERENCES;89
8;Part II: Vector and Parallel Processing;90
8.1;CHAPTER 7. SOLVING ELLIPTIC PROBLEMS ON AN
ARRAY PROCESSOR SYSTEM;92
8.1.1;I. INTRODUCTION;92
8.1.2;II. ELLIPTIC PROBLEM SOLVERS;93
8.1.3;III. VERY LARGE PROBLEMS;101
8.1.4;IV. SUMMARY;106
8.1.5;ACKNOWLEDGMENTS;106
8.1.6;REFERENCES;106
8.2;CHAPTER 8. Parallel Architectures for Iterative Methods on
Adaptive, Block Structured Grids;108
8.2.1;I. INTRODUCTION;108
8.2.2;II. DATA STRUCTURE;109
8.2.3;III. ADAPTIVE SOLUTION ALGORITHM;112
8.2.4;IV. ARCHITECTURE;113
8.2.5;V. EXPERIMENTAL PERFORMANCE AND CURRENT PROJECT STATUS;118
8.2.6;VI. REFERENCES;119
8.3;CHAPTER 9. HIGHLY CONCURRENT ALGORITHMS FOR SOLVING LINEAR SYSTEMS OF
EQUATIONS;120
8.3.1;I. INTRODUCTION;120
8.3.2;II. LU-DECOMPOSITION;123
8.3.3;III. QR-DECOMPOSITION;128
8.3.4;IV. CONJUGATE GRADIENT METHODS;129
8.3.5;V. PRECONDITIONED CONJUGATE GRADIENT METHODS;133
8.3.6;VI. CONCLUSIONS;136
8.3.7;ACKNOWLEDGMENT;137
8.3.8;REFERENCES;138
8.4;CHAPTER 10. CONJUGATE GRADIENT PRECONDITIONERS FOR
VECTOR AND PARALLEL PROCESSORS;142
8.4.1;I. INTRODUCTION;142
8.4.2;II. CONJUGATE GRADIENT ALGORITHM;143
8.4.3;Ill. PRECONDITIONERS;144
8.4.4;IV. OTHER COMPARISONS;151
8.4.5;V. CONCLUSIONS;152
8.4.6;REFERENCES;153
8.5;CHAPTER 11. AN ALGEBRA FOR SYSTOLIC COMPUTATION;156
8.5.1;1. INTRODUCTION;156
8.5.2;2. BASIC PRINCIPLES AND NOTATION—ILLUSTRATED BY A FIR FILTERING
EXAMPLE;157
8.5.3;3. FOUNDATION FOR ALGEBRAIC TRANSFORMATIONS;163
8.5.4;4. DETERMINING ALGEBRAIC TRANSFORMATIONS;168
8.5.5;5. IIR FILTERING--A FURTHER EXAMPLE;169
8.5.6;6. CONCLUDING REMARKS;171
8.5.7;ACKNOWLEDGEMENT;173
8.5.8;REFERENCES;173
8.6;CHAPTER 12. AN INCOMPLETE-CHOLESKY FACTORIZATION
BY A MATRIX PARTITION ALGORITHM;176
8.6.1;I. INTRODUCTION;176
8.6.2;II. A MATRIX-PARTITION ALGORITHM;178
8.6.3;Ill. INCOMPLETE-CHOLESKY;182
8.6.4;IV. APPLICATIONS;185
8.6.5;ACKNOWLEDGEMENTS;187
8.6.6;REFERENCES;187
8.7;CHAPTER 13. A FAST
POISSON SOLVER FOR MULTIPROCESSORS;190
8.7.1;I. INTRODUCTION;190
8.7.2;II.
THE TWO-DIMENSIONAL MD-POISSON SOLVER ON A LINEAR ARRAY OF PROCESSORS;192
8.7.3;III. THE THREE-DIMENSIONAL MD-POISSON SOLVER ON A MESH OF n2
PROCESSORS;196
8.7.4;References;200
8.8;CHAPTER 14. SYSTOLIC ARRAYS: HIGH PERFORMANCE PARALLEL
MACHINES FOR MATRIX COMPUTATION;202
8.8.1;I. INTRODUCTION;202
8.8.2;II. LU AND QR FACTORIZATIONS;203
8.8.3;III. EIGENVALUE COMPUTATION;204
8.8.4;IV. SINGULAR VALUE DECOMPOSITION
(SVD);205
8.8.5;V. DECOMPOSING ARRAYS;206
8.8.6;VI. MULTIGRID METHODS;207
8.8.7;ACKNOWLEDGEMENT;207
8.8.8;REFERENCES;208
9;Part III: Iterative Equation Solving;210
9.1;CHAPTER 15. A SURVEY OF RECENT RESULTS ON ITERATIVE METHODS
FOR SOLVING LARGE SPARSE LINEAR SYSTEMS;212
9.1.1;I. INTRODUCTION;212
9.1.2;II. A NEW IDENTITY FOR THE SSOR METHOD;213
9.1.3;III. CONVERGENCE AND DIVERGENCE DOMAINS FOR SSOR, APPLIED
TO H-MATRICES;216
9.1.4;IV. BLOCK ITERATIVE METHODS FOR SPARSE LEAST-SQUARES PROBLEMS;221
9.1.5;V. REGULAR SPLITTINGS OF MATRICES;226
9.1.6;REFERENCES;230
9.2;CHAPTER 16. NUMERICAL ALGORITHMS FOR INDEFINITE PROBLEMS;234
9.2.1;I. INTRODUCTION;234
9.2.2;II. ITERATION METHODS FOR INDEFINITE MATRICES;236
9.2.3;III. DIRECT REDUCTIONS TO POSITIVE DEFINITE PROBLEMS;240
9.2.4;REFERENCES;246
9.3;CHAPTER 17. PRECONDITIONED CONJUGATE GRADIENT METHODS
FOR THE HELMHOLTZ EQUATION;248
9.3.1;I. INTRODUCTION;248
9.3.2;II. ITERATION METHOD;250
9.3.3;III. RESULTS FOR SSOR PRECONDITIONING;252
9.3.4;IV. RESULTS FOR ADI PRECONDITIONING;255
9.3.5;REFERENCES;257
9.4;CHAPTER 18. SOLVING ELLIPTIC PROBLEMS ON
REGIONS PARTITIONED INTO SUBSTRUCTURES;260
9.4.1;I. INTRODUCTION;260
9.4.2;II. SUBSTRUCTURED PROBLEMS AND BLOCK FORM OF THE STIFFNESS
MATRICES;262
9.4.3;III. CONJUGATE GRADIENT ALGORITHMS FOR SUBSTRUCTURED
PROBLEMS AND AN INFORMAL THEORY;265
9.4.4;IV. NUMERICAL EXPERIMENTS;268
9.4.5;REFERENCES;269
9.5;CHAPTER 19. THE AD-HOC SOR METHOD: A LOCAL RELAXATION SCHEME;272
9.5.1;I. INTRODUCTION;272
9.5.2;II. THE BASIC PROCEDURE;273
9.5.3;III. BOUNDARY CONDITION AND REGION SHAPE MODIFICATIONS;274
9.5.4;IV. REAL LINEAR SYSTEMS;274
9.5.5;V. COMPLEX LINEAR SYSTEMS;277
9.5.6;VI. NONLINEAR SYSTEMS;278
9.5.7;VII. CONCLUSIONS;283
9.5.8;BIBLIOGRAPHY;283
9.6;CHAPTER 20. ITERATIVE METHODS FOR NON-SELF-ADJOINT
ELLIPTIC PROBLEMS;286
9.6.1;1. INTRODUCTION;286
9.6.2;2. KRYLOV SUBSPACE METHODS;288
9.6.3;3. A HYBRID METHOD;291
9.6.4;4. SOME PRECONDITIONING TECHNIQUES;293
9.6.5;5. NUMERICAL EXPERIMENTS;295
9.6.6;6. OPEN QUESTIONS;295
9.6.7;References;297
9.7;CHAPTER 21. IMPLICIT BLOCK EXPLICIT OVERRELAXATION SCHEMES;300
9.7.1;I. INTRODUCTION;300
9.7.2;II. GROUP ITERATIVE METHODS;301
9.7.3;III. BLOCK EXPLICIT ITERATIVE METHODS;303
9.7.4;IV. THE SOLUTION OF BOUNDARY VALUE PROBLEMS BY BLOCK EXPLICIT
ITERATIVE METHODS;307
9.7.5;V. NUMERICAL RESULTS FOR EXPLICIT BLOCK SOR METHODS;311
9.7.6;VI. IMPLICIT BLOCK EXPLICIT OVERRELAXATION SCHEMES;313
9.7.7;VII. REFERENCES;314
9.8;CHAPTER 22. DYNAMIC ACCELERATION OF NONLINEAR ITERATIONS;316
9.8.1;I. INTRODUCTION;316
9.8.2;II. ONE-STEP METHOD;318
9.8.3;III. TWO-STEP METHOD;322
9.8.4;IV. DETERMINING LINEARITY;324
9.8.5;V. TRUST REGIONS;325
9.8.6;VI. NUMERICAL RESULTS;326
9.8.7;REFERENCES;328
9.9;CHAPTER 23. NAVIER-STOKES PRESSURE EQUATION ITERATION;330
9.9.1;I. INTRODUCTION;330
9.9.2;II. ANALYSIS OF A MODEL PROBLEM;331
9.9.3;III. ADDITIVE COLUMN RENORMALIZATION;333
9.9.4;IV. SUMMARY;335
9.9.5;V. REFERENCES;336
9.10;CHAPTER 24. ACCELERATING NONSYMMETRIZABLE ITERATIVE METHODS;338
9.10.1;I. INTRODUCTION;338
9.10.2;II. BASIC ITERATIVE METHODS;339
9.10.3;III. ACCELERATION OF SYMMETRIZABLE METHODS;341
9.10.4;IV. NONSYMMETRIZABLE METHODS: THE GCW METHOD;342
9.10.5;V. NORMAL EQUATIONS;343
9.10.6;VI. CHEBYSHEV ACCELERATION IN THE NONSYMMETRIZABLE CASE;346
9.10.7;VII. GENERALIZED CG ACCELERATION;347
9.10.8;VIII. SIMPLIFICATIONS OF GENERALIZED CG ACCELERATION;350
9.10.9;IX. THE LANCZOS METHOD;353
9.10.10;REFERENCES;355
10;Part IV: Finite Elementand Multigrid Methods;358
10.1;CHAPTER 25. ADAPTIVE FINITE ELEMENT PROCESSES
IN STRUCTURAL MECHANICS;360
10.1.1;I. INTRODUCTION;360
10.1.2;II. FEEDBACK AND ADAPTIVITY;362
10.1.3;III. THE FEARS SYSTEM; EFFECTIVITY INDICES;366
10.1.4;IV. OTHER VERSIONS OF THE FINITE ELEMENT METHOD;371
10.1.5;V. POST-PROCESSING;374
10.1.6;VI. PROBLEM FORMULATIONS IN THE NONLINEAR CASE;376
10.1.7;VII.
ERROR ESTIMATES;379
10.1.8;VIII. CONTINUATION PROCESSES;384
10.1.9;IX. REFERENCES;389
10.2;CHAPTER 26. FINITE ELEMENT APPROXIMATIONS TO
COMPRESSIBLE FLOW PROBLEMS;394
10.2.1;I. INTRODUCTION;394
10.2.2;II. INTRINSIC INSTABILITIES IN GALERKIN FORMULATIONS;395
10.2.3;III. LEAST SQUARES METHODS;400
10.2.4;IV. THE PHOENIX EFFECT;401
10.2.5;V. CONCLUSIONS;405
10.2.6;REFERENCES;406
10.3;CHAPTER 27. SOLVING ELLIPTIC PROBLEMS BY DOMAIN DECOMPOSITION
METHODS WTIH APPLICATIONS;410
10.3.1;I. INTRODUCTION;410
10.3.2;II. FORMULATION OF THE MODEL PROBLEM. GENERALITIES;410
10.3.3;III. THE SCHWARZ ALTERNATING METHOD;413
10.3.4;IV. DOMAIN DECOMPOSITION METHODS WITH LAGRANGE MULTIPLIERS;414
10.3.5;V. A NEW DOMAIN DECOMPOSITION METHOD;424
10.3.6;VI. NUMERICAL EXPERIMENTS;427
10.3.7;VII. APPLICATION TO THE NUMERICAL SIMULATION OF TRANSONIC
FLOWS ON LARGE COMPUTATIONAL DOMAINS;428
10.3.8;VIII. FURTHER COMMENTS. CONCLUSION;439
10.3.9;REFERENCES;440
10.4;CHAPTER 28. COMPUTING ADDED MASS COEFFICIENTS;442
10.4.1;I. PRELIMINARY REMARKS;442
10.4.2;II. HIGHER ANALYSIS;443
10.4.3;III. APPLYING ELLPACK 1978 TO CYLINDERS;445
10.4.4;IV. NETWORK ANALOGY;446
10.4.5;V. DIPOLE MOMENT METHOD;448
10.4.6;VI. ADDED MASS OF SOLIDS;449
10.4.7;REFERENCES;451
10.5;CHAPTER 29. A PRIORI LOCAL GRID REFINEMENT IN THE MULTIGRID METHOD;454
10.5.1;References;466
10.6;CHAPTER 30. ABSTRACT MULTI-GRID WITH APPLICATIONS TO ELLIPTIC
BOUNDARY-VALUE PROBLEMS;468
10.6.1;I. GENERAL THEORY;468
10.6.2;II. ELLIPTIC BOUNDARY-VALUE PROBLEMS EXAMPLES;474
10.6.3;ACKNOWLEDGEMENTS;480
10.6.4;REFERENCES;480
10.7;CHAPTER 31. A NEW ORDERING SCHEME FOR THE HERMITE BICUBIC
COLLOCATION EQUATIONS;482
10.7.1;I. INTRODUCTION;482
10.7.2;II. THE COLLORDER ORDERING;484
10.7.3;III. PIVOTING IN GAUSS ELIMINATION WITH THE COLLORDER ORDERING;487
10.7.4;IV. WHY PIVOTING IS NOT NEEDED WITH THE COLLORDER ORDERING;491
10.7.5;V. COMPUTATIONAL EFFICIENCY;493
10.7.6;VI. CONCLUSIONS AND CONJECTURES;493
10.7.7;REFERENCES;494
10.8;CHAPTER 32. SOLUTION METHODS FOR THE FINITE ELEMENT EQUATIONS IN
NONLINEAR SHELL ANALYSIS;496
10.8.1;References;504
11;Part V: Advances in Modelingand Physical Applications;506
11.1;CHAPTER 33. DISCRETIZATION AND MULTILEVEL SOLUTION
TECHNIQUES FOR NONLINEAR ELLIPTIC SYSTEMS;508
11.1.1;1. INTRODUCTION;508
11.1.2;2. SCALING;509
11.1.3;3. DISCRETIZATION;511
11.1.4;4. SEMICONDUCTOR DEVICE MODELS;515
11.1.5;5. MULTI-LEVEL TECHNIQUES;517
11.1.6;REFERENCES;520
11.2;CHAPTER 34. ANALYSIS OF NONLINEAR ELLIPTIC SYSTEM
SARISING IN REACTION/DIFFUSION MODELING;522
11.2.1;I. INTRODUCTION;522
11.2.2;II. MODEL DEVELOPMENT;524
11.2.3;III. MAXIMUM PRINCIPLES FOR THE SEMICONDUCTOR SYSTEM;527
11.2.4;IV. INVARIANT RECTANGLES FOR CHEMICAL SYSTEMS;529
11.2.5;V. COMPACTNESS AND CONTINUITY OF THE SCHAUDER MAP: EXISTENCE;531
11.2.6;VI. A QUASI-CONSTRUCTIVE FRAMEWORK; PSEUDOMONOTONE OPERATORS;532
11.2.7;REFERENCES;534
11.3;CHAPTER 35. NUMERICAL METHODS FOR SOLVING COUPLED SEMICONDUCTOR EQUATIONS
ON A MINICOMPUTER;536
11.3.1;I. INTRODUCTION;536
11.3.2;II. THE MATHEMATICAL MODEL;537
11.3.3;III. SPECIAL FEATURES OF THE SEMICONDUCTOR DEVICE EQUATIONS;538
11.3.4;IV. SELECTED DISCUSSION OF CHOICES;543
11.3.5;V. GRAPHICS;545
11.3.6;VI. REFERENCES;545
11.4;Chapter 36. Numerical Solution of a Model Equation Near the Onset of
the Rayleigh-Benard Instability;546
11.4.1;1. Introduction;546
11.4.2;2. A Two-Dimensional Equation for Three-Dimensional Convection;548
11.4.3;3. A Model System with Vorticity;553
11.4.4;4. Conclusions;556
11.4.5;References;557
11.5;CHAPTER 37. APPROXIMATION OF A NONSYMMETRIC,
SINGULARLY PERTURBED PROBLEM;560
11.5.1;I. INTRODUCTION;560
11.5.2;II. A HODIE SCHEME;561
11.5.3;III. INNER/OUTER DECOMPOSITION;564
11.5.4;IV. NUMERICAL RESULTS AND CONCLUSIONS;568
11.5.5;REFERENCES;570
11.6;CHAPTER 38. NUMERICAL SOLUTION OF A CONFINED LAMINAR
DIFFUSION FLAME;572
11.6.1;I. INTRODUCTION;572
11.6.2;II. PROBLEM FORMULATION;573
11.6.3;III. METHOD OF SOLUTION;578
11.6.4;IV. NUMERICAL RESULTS;582
11.6.5;REFERENCES;583
12;Index;584