Numerical Solution of Partial Differential Equations-II, Synspade 1970 (eBook)

Proceedings of the Second Symposium on the Numerical Solution of Partial Differential Equations, SYNSPADE 1970, Held at the University of Maryland, College Park, Maryland, May 11-15, 1970

Bert Hubbard (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
658 Seiten
Elsevier Science (Verlag)
978-1-4832-6248-2 (ISBN)

Infos

Numerical Solution of Partial Differential Equations-II: Synspade 1970 provides information pertinent to the fundamental aspects of partial differential equations. This book covers a variety of topics that range from mathematical numerical analysis to numerical methods applied to problems in mechanics, meteorology, and fluid dynamics. Organized into 18 chapters, this book begins with an overview of the methods of the Rayleigh-Ritz-Galerkin type for the approximation of boundary value problems using spline basis functions and Sobolev spaces. This text then analyzes a special approach aimed at solving elliptical equations. Other chapters consider the approximation theoretic study of special sets of approximating functions. This book discusses as well combining the alternating-direction methods with Galerkin methods to obtain highly efficient procedures for the numerical solution of second order parabolic and hyperbolic problems. The final chapter deals with the results concerning Chebyshev rational approximations of reciprocals of certain entire functions. This book is a valuable resource for mathematicians.

Inhalt

Front Cover 1
Numerical Solution of Partial Differential Equations-II: Synspade 1970 4
Copyright Page 5
Table of Contents 6
CONTRIBUTORS 8
PREFACE 10
CHAPTER 1. SOME ASPECTS OF THE METHOD OF THE HYPERCIRCLE APPLIED TO ELLIPTIC VARIATIONAL PROBLEMS 12
Introduction 12
1. Example of Conjugate Problems and a posteriori Estimates 16
2. A posteriori Estimates and Conjugate Problems of Abstract Boundary Value Problems 27
3. Approximations of the Spaces H.m(O,D*) and H.m(O,D*) 50
4. Approximation of Conjugate Boundary Value Problems 64
REFERENCES 77
CHAPTER 2. THE FINITE ELEMENT METHOD FOR ELLIPTIC DIFFERENTIAL EQUATIONS 80
1. Introduction 80
2. Approximation Theory in Rn 81
3. Boundary Value Problems with Natural Boundary Conditions 86
4. Boundary Value Problem with Natural Boundary Conditions and Singularities on the Boundary 98
5. Boundary Value Problem with Dirichlet Boundary Conditions-- The Case of the Lipschitz Domain 101
6. The Solution of the Dirichlet Problem by the Perturbed Variational Principle 104
7. Computational Aspects and Numerical Experiments 106
REFERENCES 109
CHAPTER 3. ON THE NUMERICAL SOLUTION OF ELLIPTIC BOUNDARY VALUE PROBLEMS BY LEAST SQUARES APPROXIMATION OF THE DATA 118
1. Introduction 118
2. Preliminaries 120
3. Finite Dimensional Subspaces of Hk(R) 124
4. Least Squares Methods for 2mth Order Boundary Value Problems 125
5. Interface Problems - Equations with Discontinuous Coefficients 130
6. Examples 135
REFERENCES 140
CHAPTER 4. ALTERNATING-DIRECTION GALERKIN METHODS ON RECTANGLES 144
1. Introduction 144
2. An Alternating-Direction Galerkin Method for the Heat Equation on a Rectangle 149
3. General Parabolic Equations 164
4. Hyperbolic Problems 194
5. Elliptic Problems 202
REFERENCES 223
CHAPTER 5. ON THE DIFFERENCE EQUATIONS FOR METEOROLOGICAL PREDICTION 226
1. Introduction 227
2. The "Empirical" Equations of Meteorological Prediction 233
3. Development of a Set of Complete Difference Equations 238
REFERENCES 251
CHAPTER 6. FURTHER DEVELOPMENTS IN THE APPROXIMATION THEORY OF EIGENVALUES 254
1. Eigenvalue Problems for Linear Elliptic Systems 255
2 . Upper Approximation of 261
REFERENCES 263
CHAPTER 7. NUMERICAL DESIGN OF TRANSONIC AIRFOILS 264
1. Introduction 264
2. Transonic Controversy and the Inverse Problem 266
3. The Method of Complex Characteristics 268
4. Continuation Around the Sonic Locus 275
5. Construction and Properties of the Airfoils 278
REFERENCES 279
CHAPTER 8. ON THE NUMERICAL TREATMENT OF PARTIAL DIFFERENTIAL EQUATIONS BY FUNCTION THEORETIC METHODS 284
1. Introduction 284
2. Boundary Value Problems: Analytic Methods 285
3. Boundary Value Problems: Numerical Treatment 305
4 . Nonlinear Equations in Two Independent Variables: The Dirichlet Problem 323
5. Nonlinear Equations in Two Independent Variables: The Cauchy Problem 328
REFERENCES 334
CHAPTER 9. A NEW DIFFERENCE SCHEME FOR PARABOLIC PROBLEMS 338
1. Introduction 338
2. The Box Scheme 340
3. Error Estimates 343
4. Richardson Extrapolation 348
5. Solution of the Difference Equations 353
REFERENCES 361
CHAPTER 10. SINGULARITIES IN INTERFACE PROBLEMS 362
1. Introduction 362
2. A Density Theorem in Two Dimensions 363
3. The Method of Continuation 372
4. The General Equation in the Plane 379
5. A Density Theorem in Three Dimensions 386
6. The Interface Problem in a Polyhedron 406
REFERENCES 411
CHAPTER 11. INITIAL BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL AND DIFFERENCE EQUATIONS IN ONE SPACE DIMENSION 412
1. Differential Equations 412
2. Difference Approximations for Hyperbolic Systems 420
REFERENCES 427
CHAPTER 12. NUMERICAL SOLUTION OF CONDITIONALLY PROPERLY POSED PROBLEMS 428
REFERENCES 443
CHAPTER 13. ON GENERAL PURPOSE PROGRAMS FOR FINITE ELEMENT ANALYSIS, WITH SPECIAL REFERENCE TO GEOMETRIC AND MATERIAL NONLINEARITIES 444
Abstract 444
Introduction 445
Review of Literature 446
Technical Considerations 450
General Purpose Programs 454
Note on the Cost of Computing 458
Case Studies 460
1. Substructural Analysis of the 747 Aircraft Wing-Body Intersection 461
2. Elastic-Plastic Analysis of Tensile Specimen with Semi-Elliptic Crack 462
3. Analysis of Shell-Nozzle Junction with Combined Shell and Triangular Ring Elements 462
4. Imperfect Hemisphere under External Pressure 463
5. Infinite Incompressible Log 465
Conclusions and Future Work 465
REFERENCES 466
CHAPTER 14. ON THE THEORY OF THE SPLITTING-UP METHOD 480
1. A Scheme of Splitting Up of Nonstationary Problems 480
2. Hydrodynamic Transfer Equations of Transfer Along Trajectories 490
3. Solution of Time-Independent Problems by the Splitting Up Method with Variational Optimization 500
REFERENCES 510
CHAPTER 15. ON CLASSES OF n-DIMENSIONAL NONLINEAR MAPPINGS GENERALIZING SEVERAL TYPES OF MATRICES 512
1. Introduction 512
2. Basic Definitions 515
3. Properties of the Different Functions 520
4. Surjectivity 529
5. Boundary Value Problems 539
6. Iterative Processes 544
REFERENCES 555
CHAPTER 16. THE FINITE ELEMENT METHOD AND APPROXIMATION THEORY 558
1. Approximation 567
2. Stability 581
3. Error Estimates 588
REFERENCES 593
CHAPTER 17. ON THE RATE OF CONVERGENCE OF DIFFERENCE SCHEMES FOR HYPERBOLIC EQUATIONS 596
1. Introduction 596
2. Function Spaces and Interpolation 598
3. The Rate of Convergence in the Stable Cases 602
4. Maximum-Norm Estimates for L2-Stable Operators 606
5. Lp-Estimates in the Non-Stable Scalar One-Dimensional Constant Coefficient Case 610
6. Isolated Singularities 614
7. Propagation of Discontinuities in the Constant Coefficient Case 618
8. Propagation of Discontinuities in the Variable Coefficient Case 622
REFERENCES 632
CHAPTER 18. SOME RESULTS IN APPROXIMATION THEORY WITH APPLICATIONS TO NUMERICAL ANALYSIS 634
1. Introduction 634
2. Chebyshev Semi-Discrete Approximations of Parabolic Partial Difference Equations 635
3. Theoretical Extensions 639
4. Numerical Results 645
5. Improved Error Bounds for Spline and L-Spline Interpolation 648
6. Application to Two-Point Boundary Value Problems 658
REFERENCES 659

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 1-4832-6248-0 / 1483262480
• ISBN-13: 978-1-4832-6248-2 / 9781483262482