Front Cover 1
Elliptic Problem Solvers 4
Copyright Page 5
Table of Contents 6
Contributors 10
Preface 14
CHAPTER 1. A MULTI-LEVEL ITERATIVE METHOD FOR NONLINEAR ELLIPTIC EQUATIONS 16
I. INTRODUCTION 16
II. A MULTI-LEVEL ITERATIVE METHOD FOR LINEAR PROBLEMS 17
III. ADAPTIVE REFINEMENT 21
IV. A MULTI-LEVEL ITERATIVE METHOD FOR NONLINEAR PROBLEMS 22
V. NUMERICAL EXAMPLE 24
REFERENCES 30
CHAPTER 2. SOLVING ELLIPTIC PROBLEMS: 1930-1980 32
1. THE STATE OF KNOWLEDGE IN 1930 32
2. Three Significant Advances: 1930-1945 34
3. Iterative Methods: 1947-1962 36
4. Chebyshev Methods 38
5. Finite Element Methods 41
6. Some Intriguing Mathematical Questions 43
7. Direct sparse matrix methods 47
8. Current trends 49
REFERENCES 51
CHAPTER 3. MULTIGRID SOLVERS ON PARALLEL COMPUTERS 54
1. INTRODUCTION 55
2. BASIC PROCESSES 57
3. PARALLEL MULTIGRID PROCESSING 70
ACKNOWLEDGMENTS 96
REFERENCES 96
CHAPTER 4. IMPLEMENTING TECHNIQUES FOR ELLIPTIC PROBLEMS ON VECTOR PROCESSORS 100
I. INTRODUCTION 100
II. VECTOR PROCESSORS 101
III. IMPLEMENTING TECHNIQUES FOR ELLIPTIC PROBLEMS 107
IV. RESEARCH ISSUES 109
REFERENCES 110
CHAPTER 5. ON SOME TRENDS IN ELLIPTIC PROBLEM SOLVERS 114
1. Introduction 114
2. Architecture 115
3. Algorithms 118
REFERENCES 128
CHAPTER 6. CO-ENERGY METHODS FOR ELLIPTIC FLOW AND RELATED PROBLEMS 130
1. INTRODUCTION 130
2. DIRECT AND COMPLEMENTARY FORMULATIONS 131
3. APPROXIMATION AND ERROR ANALYSIS 136
4. DISCRETE SYSTEM ALGORITHMS 144
ACKNOWLEDGMENTS 148
REFERENCES 148
CHAPTER 7. ELLPACK: PROGRESS AND PLANS 150
I. INTRODUCTION 150
II. THE ELLPACK SYSTEM AND MODULES 151
III. PERFORMANCE EVALUATION METHODOLOGY 156
IV. PERFORMANCE EVALUATIONS 159
IV. PLANS FOR A "CORE" ELLPACK 168
VI. THE FUTURE ELLPACK STRUCTURE 169
VII. FUTURE PERFORMANCE EVALUATION PLANS 173
VIII. NON RESEARCH APPLICATIONS OF ELLPACK 174
REFERENCES 175
CHAPTER 8. THE ITPACK PACKAGE FOR LARGE SPARSE LINEAR SYSTEMS 178
I. INTRODUCTION AND BACKGROUND 178
II. ITERATIVE ALGORITHMS: BASIC ITERATIVE METHODS 180
III. ACCELERATION PROCEDURES 183
IV. STOPPING PROCEDURES 186
V. SPARSE STORAGE FORMAT 188
VI. PROGRAMS OF ITPACK 189
VII. THE USE OF ITPACK WITH ELLPACK 190
VIII. NUMERICAL EXPERIMENTS 191
IX. FUTURE PLANS 194
X. ITPACK INFORMATION 196
APPENDIX: ORTHORES: A GENERALIZED CONJUGATE GRADIENT ACCELERATION PROCEDURE 196
REFERENCES 198
CHAPTER 9. EFFICIENT FORTRAN SUBPROGRAMS FOR THE SOLUTION OF ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 202
I. INTRODUCTION 202
II. THE PACKAGE 202
REFERENCES 205
CHAPTER 10. ITERATIVE METHODS FOR FINITE ELEMENT EQUATIONS 206
1. INTRODUCTION 206
2. SOME PRECONDITIONING TECHNIQUES, IN PARTICULAR FOR FINITE ELEMENT EQUATIONS 207
REFERENCES 210
CHAPTER 11. PREDICTOR-CORRECTOR METHODS FOR THE SOLUTION OF TIME-DEPENDENT PARABOLIC PROBLEMS ON PARALLEL PROCESSORS 212
I. INTRODUCTION 212
II. STABILITY ANALYSIS 213
III. EXPERIMENTS 214
REFERENCES 216
NOTICE 216
CHAPTER 12. EFFICIENT SOLUTION OF THE BIHARMONIC EQUATION 218
I. INTRODUCTION 219
II. OUTLINE OF THEORY 221
III. COMPUTER IMPLEMENTATIONS 224
IV. EXTENSIONS AND APPLICATIONS 226
ACKNOWLEDGMENTS 228
REFERENCES 231
CHAPTER 13. ATTAINABLE ACCURACY OF COMPACT DISCRETIZATIONS OF THE POISSON EQUATION 234
I. INTRODUCTION 234
II. OPTIMALITY RESULTS 235
III. THE AFFECT ON GLOBAL ERROR 237
REFERENCES 238
CHAPTER 14. THE CONCEPT OF RIGIDITY AND ITS IMPLEMENTATION 240
I. INTRODUCTION 240
II. CONCEPTS AND ASSOCIATED TYPES 241
III. DESIGN PRINCIPLES AND SCOPE 242
REFERENCES 244
CHAPTER 15. THEOREMS OF STEIN-ROSENBERG TYPE II. OPTIMAL PATHS OF RELAXATION IN THE COMPLEX PLANE 246
1. INTRODUCTION 246
2. OPTIMAL PATHS OF RELAXATION FOR Jw 248
3. OPTIMAL PATHS OF RELAXATION FOR Lw 251
References 254
CHAPTER 16. SPARSE VECTORIZED DIRECT SOLUTION OF ELLIPTIC PROBLEMS 256
I. INTRODUCTION 256
II. SPARSE SOLUTION OF ELLIPTIC PROBLEMS 256
ACKNOWLEDGMENT 260
REFERENCES 260
CHAPTER 17. MULTI-GRID AND ICCG FOR PROBLEMS WITH INTERFACES 262
Acknowledgments 267
ADDENDUM 268
CHAPTER 18. AN AD HOC SIR METHOD 270
REFERENCE 274
CHAPTER 19. ON PRECONDITIONED ITERATIVE METHODS FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 276
1. INTRODUCTION 276
2. SPARSE PRECONDITIONING METHODS 278
3. COMPACT PRECONDITIONING METHODS 281
REFERENCES 283
CHAPTER 20. BLOCK RELAXATION STRATEGIES 286
I. INTRODUCTION 286
II. STEPS 287
III. PARTICULAR STRATEGIES TEST PROBLEMS
IV. RESULTS 289
V. CONCLUSIONS 290
REFERENCES 290
CHAPTER 21. ON THE NUMERICAL SOLUTION OF NONLINEAR ELLIPTIC PDEs ARISING FROM SEMICONDUCTOR DEVICE MODELING 292
I. INTRODUCTION AND EQUATION FORMULATION 292
II. DISCRETE EQUATION SOLUTION 294
III. EXAMPLES 297
REFERENCES 299
CHAPTER 22. NON-STANDARD MULTIGRID TECHNIQUES USING CHECKERED RELAXATION AND INTERMEDIATE GRIDS 300
1. INTRODUCTION 300
2. THEORETICAL RESULTS FOR THE DIRICHLET MODEL PROBLEM 302
3. NUMERICAL EXPERIMENTS FOR THE NEUMANN MODEL PROBLEM (HELMHOLTZ EQUATION) 308
4. THE DIRICHLET PROBLEM FOR GENERAL DOMAINS 310
5. ADDITIONAL REMARKS 313
REFERENCES 314
CHAPTER 23. SOME EXPERIMENTS IN SOLVING STIFF OSCILLATORY ORDINARY DIFFERENTIAL EQUATIONS 316
SUMMARY 316
REFERENCES 319
CHAPTER 24. A NUMERICAL METHOD FOR SOLVING ELLIPTIC BOUNDARY VALUE PROBLEMS IN UNBOUNDED DOMAINS 322
l. INTRODUCTION 322
II. THE MESH GRADING PROCEDURE 323
III. ERROR ESTIMATES 325
IV. NUMERICAL RESULTS 326
REFERENCES 328
CHAPTER 25. APPLICATIONS OF TRANSFINITE ("BLENDING-FUNCTION") INTERPOLATION TO THE APPROXIMATE SOLUTION OF ELLIPTIC PROBLEMS 330
1. INTRODUCTION 330
2. EXACT MATCHING OF BOUNDARY CONDITIONS 332
3. HOMOGENIZATION OF BOUNDARY CONDITIONS 340
4. INCORPORATION OF AUXILIARY CONSTRAINTS 343
5. COLLOCATION USING TRANSFINITE INTERPOLATION FORMULAS 348
6. TIME-DEPENDENT AND RELATED PROBLEMS 350
ACKNOWLEDGMENTS 351
REFERENCES 352
CHAPTER 26. APPLICATION OF A PARALLEL PROCESSOR TO THE SOLUTION OF FINITE DIFFERENCE PROBLEMS 354
l. INTRODUCTION TO DAP 354
II. APPLICATION TO FIELD PROBLEMS 355
III. TECHNIQUES FOR IMPLICIT METHODS 356
IV. THE EXAMPLE PROBLEM 357
V. RESULTS 358
VI. CONCLUSIONS 359
REFERENCES 359
CHAPTER 27. VECTOR ALGORITHMS FOR ELLIPTSC PARTIAL DIFFERENTIAL EQUATIONS BASED ON THE JACOBI METHOD 360
I. INTRODUCTION 360
II. CONJUGATE GRADIENT JACOBI METHODS 362
III. NUMERIC RESULTS 364
REFERENCES 366
CHAPTER 28. ADAPTING ITERATIVE ALGORITHMS DEVELOPED FOR SYMMETRIC SYSTEMS TO NONSYMMETRIC SYSTEMS 368
I. INTRODUCTION 368
II. MOTIVATION OF MODIFIED FORMULAS FOR CONJUGATE GRADIENT ACCELERATION 369
III. MODEL PROBLEM AND CENTRAL DIFFERENCE DISCRETIZATION 370
IV. MODIFIED UPWIND DIFFERENCE REPRESENTATION 372
V. MODIFIED CENTRAL DIFFERENCE DISCRETIZATION 372
ACKNOWLEDGEMENTS 374
REFERENCES 374
CHAPTER 29. COMPARISON OF METHODS OF SOLUTION OF THE FINITE ELEMENT EQUATIONS FOR THE LARGE DISPLACEMENT ANALYSIS OF ARCHES 376
References 382
CHAPTER 30. MESH GENERATION BY CONFORMAL AND QUASICONFORMAL MAPPINGS 384
I. INTRODUCTION 384
II. BOUNDARY VALUE PROBLEM FOR QUASICONFORMAL MAPPINGS 385
III. EXAMPLES 386
IV. CONCLUSIONS 388
REFERENCES 388
CHAPTER 31. BLOCK ITERATIVE METHODS 390
I. INTRODUCTION 390
II. THE MODEL PROBLEM 391
III. A MODIFIED 3-d PROBLEM 392
IV. FINITE ELEMENT PROBLEMS 394
V. FURTHER RESEARCH 396
REFERENCES 396
CHAPTER 32. A MESH - PARAMETER - CONTINUATION METHOD 398
I. INTRODUCTION 398
II. SEMICONDUCTOR PROBLEMS 399
III. THE PCC METHOD 400
IV. PARAMETER - CONTINUATION (PC) 401
V. MESH - CONTINUATION 402
VI. MESH - PARAMETER - CONTINUATION (MPC) 403
VII. 2-D MESHES 403
REFERENCES 404
CHAPTER 33. CAPACITANCE MATRIX METHODS - A BRIEF SURVEY 406
ACKNOWLEDGEMENTS 412
REFERENCES 412
CHAPTER 34. GEM SOLUTIONS OF ELLIPTIC AND MIXED PROBLEMS WITH NON-SEPARABLE 5- AND 9-POINT OPERATORS 414
INTRODUCTION 414
PROBLEM DESCRIPTION IN THE GEM CODE 415
TEST PROBLEMS AND RESULTS 417
REFERENCES 418
CHAPTER 35. A PARALLEL BLOCK STIEFEL METHOD FOR SOLVING POSITIVE DEFINITE SYSTEMS 420
I. INTRODUCTION 420
II. THE BLOCK-STIEFEL ITERATION (B-St.) 421
III. THE BLOCK CONJUGATE GRADIENT METHOD 424
IV. SOME REMARKS AND NUMERICAL EXPERIMENTS 424
REFERENCES 426
CHAPTER 36. Numerical Solution of Coupled Systems of Partial Differential Equations in One Spatial Variable and Time 428
1. Introduction 428
2. The Extension 429
3. Examples 429
4. Two Dimensions 430
Bibliography 432
CHAPTER 37. ON THE CHOICE OF DISCRETlZATION FOR SOLVING P.D.E. 'S ON A MULTI-PROCESSOR 434
ACKNOWLEDGEMENT 437
REFERENCES 437
CHAPTER 38. A SOFTWARE PACKAGE FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 438
1. INTRODUCTION 438
2. ORGANIZATION OF THE PACKAGE 439
REFERENCES 442
CHAPTER 39. AN EMPIRICAL INVESTIGATION OF METHODS FOR NONSYMMETRIC LINEAR SYSTEMS 444
I. INTRODUCTION 444
II. DISCRETIZATIONS 445
III. SOLUTION METHODS 446
IV. COMPARISONS AND CONCLUSIONS 447
REFERENCES 448
CHAPTER 40. SEMICONDUCTOR DEVICE SIMULATION 450
Acknowlegments 454
References 454
Index 456