Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (eBook)

Front Cover 1
The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations 4
Copyright Page 5
Table of Contents 6
CONTRIBUTORS 10
PREFACE 14
PART I: SURVEY LECTURES ON THE MATHEMATICAL FOUNDATIONS OF THE FINITE ELEMENT METHOD 18
FOREWORD 20
CHAPTER 1. PRELIMINARY REMARKS 22
1.1. Introduction 22
1.2. Numerical solution of partial differential equations 24
1.3. Finite Element Method. 25
1.4. The sources of the theory of the finite element method 27
1.5. The mathematical foundations of the finite element method. 28
Reference 30
CHAPTER 2. THE FUNDAMENTAL NOTIONS 32
2.1. Introduction. 32
2.2 Domains 32
2.3. Sobolev spaces Hl (O) 34
2.4. Sobolev spaces Ha (O.) 47
2.5. The n-width 50
2.6. The weighted Sobolev spaces 53
References 61
CHAPTER 3. PROPERTIES OF SOLUTIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS 64
Introduction 64
3.1. Strongly elliptic boundary value problems 64
3.2. Examples of elliptic operators 66
3.3. Boundary conditions 68
3.4. Examples of elliptic boundary value problems 70
3.5. The adjoint problem 74
3.6. Solvability and regularity theorems. 76
3.7. Another family of spaces 81
3.8. Solvability and regularity theorems (continued) 88
3.9. Additional results. 95
References 97
CHAPTER 4. THEORY OF APPROXIMATION 100
4.1. The (t,k) - system Sth,k (O) 100
4.2. Simultaneous approximations by (t,k)-systems 118
References. 125
CHAPTER 5. VARIATIONAL PRINCIPLES 128
5.1. Introduction. 128
5.2 The variational principle 128
5.3. Variational Principles for Second Order Differencial Equations 134
5.4. Variational principle for second order systems. 182
References 200
CHAPTER 6. RATE OF CONVERGENCE OF THE FINITE ELEMENT METHOD 202
6.1. Introduction 202
6.2. Approximation of the variational principle 202
6.3. Finite element method for the second order differential equations 206
6.4. Lower bounds for the finite element method 247
REFERENCES 256
CHAPTER 7. ONE PARAMETER FAMILIES OF VARIATIONAL PRINCIPLES 260
7-1. Introduction 260
7.2. The Penalty Method 261
7.3. The weighted least square method. 271
References. 279
CHAPTER 8, FINITE ELEMENT METHOD FOR NON-SMOOTH DOMAINS AND COEFFICIENTS 282
8.1. Introduction 282
8.2. Problems with Lipschitzian domain. 283
8.3. Problems with piecewise smooth domain 287
8.4. The problem with a piecewise smooth domain—Continuation 294
8.5. The interface problem 297
8.6. Abrupt changes of the boundary conditions 298
REFERENCES 298
CHAPTER 9. THE PROBLEMS OF PERTURBATIONS IN THE FINITE ELEMENT METHOD 302
9.1. Introduction 302
9.2. The problem of linear perturbation operators 306
9.3. Penalty method with linear perturbations 309
9.4. Problems of nonlinear perturbations 313
REFERENCES 317
CHAPTER 10. THE EIGENVALUE PROBLEM 320
10.1 Introduction 320
10.2. The eigenvalue problem 322
10.3. The linear eigenvalue problem 336
10.4. Associated Eigenvalue Problems 340
10.5. The approximation of the eigenvalue problem 343
10.6. The approximation of the eigenvalue problem (continuation) 354
10.7. Examples 355
10.8. Additional comments 357
REFERENCES. 359
CHAPTER 11. THE FINITE ELEMENT METHOD FOR TIME DEPENDENT PROBLEMS 362
11.1. Introduction. 362
11.2. The variational principle 364
11.3. The finite element method 371
REFERENCES 375
PART III: NVITED HOUR LECTURES 378
CHAPTER 12. PIECEWISE ANALYTIC INTERPOLATION AND APPROXIMATION IN TRIANGULATED POLYGONS* 380
1. Introduction 380
2. Piecewise linear interpolation 383
3. Higher-order continuous "Lagrange" interpolants 386
4 . Cubic " Hermite" approximation 388
5. Continuously differentiable interpolants 390
6. Tricubic polynomial interpolation 395
7. Right triangles 399
REFERENCES 401
CHAPTER 13. APPROXIMATION OF STEKLOV EIGENVALUES OF NON-SELFADJOINT SECOND ORDER ELLIPTIC OPERATORS 404
1. Introduction 404
2. Preliminaries 406
3. Convergence estimates 408
4. The Steklov eigenvalue problem 412
5. On the structure of T,h 422
REFERENCES 424
CHAPTER 14. THE COMBINED EFFECT OF CURVED BOUNDARIES AND NUMERICAL INTEGRATION IN ISOPARAMETRIC FINITE ELEMENT METHODS 426
1. Introduction and preliminaries 427
2. Isoparametric finite elements 435
3. "Isoparametric" numerical integration 451
4. The discrete problem: error bound in H1( (Oh). 467
5. Error bound in L2 (Oh) 480
REFERENCES 489
CHAPTER 15. A SUPERCONVERGENCE RESULT FOR THE APPROXIMATE SOLUTION OF THE HEAT EQUATION BY A COLLOCATION METHOD 492
1. Introduction 492
2. Error estimates 495
REFERENCES 506
CHAPTER 16. SOME L2 ERROR ESTIMATES FOR PARABOLIC GALERKIN METHODS 508
1. Introduction 508
2. Parabolic Regularity 514
3. Error Estimates 516
REFERENCES 521
CHAPTER 17. COMPUTATIONAL ASPECTS OF THE FINITE ELEMENT METHOD 522
1. Introduction 522
2. The asymptotic rate of convergence 527
3. Operation counts 532
4. Numerical results 536
REFERENCES 539
CHAPTER 18. EFFECTS OF QUADRATURE ERRORS IN FINITE ELEMENT APPROXIMATION OF STEADY STATE, EIGENVALUE AND PARABOLIC PROBLEMS 542
1. Introduction 542
2. A simple example 543
3. Steady state problems 552
4. The Eigenvalue Problem 559
5. Parabolic problems 565
REFERENCES 572
CHAPTER 19. EXPERIENCE WITH THE PATCH TEST FOR CONVERGENCE OF FINITE ELEMENTS 574
Introduction 574
Origins of the patch test 575
Mixability and the primitive plate bending elements 578
Eigenvalues and mixability 581
The quartic plate bending triangle 582
Properties of the isoparametric elements 585
Mixoparametric elements with Legendre polynomials 589
Ahmad shell constrained at Gauss points 595
Boundary conditions for polynomial spline solutions 600
Conclusions 601
REFERENCES 602
CHAPTER 20. HIGHER ORDER SINGULARITIES FOR INTERFACE PROBLEMS 606
1. Introduction 606
2. The principal result 607
3. The singular functions 611
4. Some interpolation spaces 613
5. The exceptional indices 615
REFERENCES 619
CHAPTER 21. ON DIRICHLET PROBLEMS USING SUBSPACES WITH NEARLY ZERO BOUNDARY CONDITIONS 620
1. Introduction 620
2. Notations, approximating subspaces 621
3. Projection methods 624
4· Cubic spline functions 632
5. Proof of the lemma 639
REFERENCES 643
CHAPTER 22. GENERALIZED CONJUGATE FUNCTIONS FOR MIXED FINITE ELEMENT APPROXIMATIONS OF BOUNDARY VALUE PROBLEMS 646
1. Introduction. 646
2. Conjugate Projections and Approximations 648
3. Finite element approximations 660
4. Complementary Variational Principles and the Hypercircle Method 671
5. Energy Error Estimates and Convergence 679
Acknowledgment 685
REFERENCES 685
CHAPTER 23. FINITE ELEMENT FORMULATION BY VARIATIONAL PRINCIPLES WITH RELAXED CONTINUITY REQUIREMENTS 688
Introduction. 688
Finite element formulation for harmonic equation 689
Finite element formulation for bi-harmonic equation 697
Remarks 702
REFERENCES 702
CHAPTER 24. VARIATIONAL CRIMES IN THE FINITE ELEMENT METHOD 706
1. Introduction 706
2. Nonconforming elements 712
3. Violation of essential boundary conditions 718
4. Numerical integration 720
REFERENCES 726
CHAPTER 25. SPLINE APPROXIMATION AND DIFFERENCE SCHEMES FOR THE HEAT EQUATION 728
1. Introduction 728
2. Some Special Trigonometric Polynomials 733
3. Spline Interpolation 738
4. The Continuous Time Galerkin Method 743
5. Application of the Finite Difference Theory 745
6. Smoothing and Best L2-Approximation of Initial Data 749
7. Discrete Time Galerkin Methods 753
8. A Numerical Example 759
REFERENCES 762
PART III: SHORT COMMUNICATIONS 764
CHAPTER 26. THE EXTENSION AND APPLICATION OF SARD KERNEL THEOREMS TO COMPUTE FINITE ELEMENT ERROR BOUNDS 766
1. Introduction 766
2. Boundary value problem and interpolation remainder theory 766
3. Sard kernel theorems applied to piecewise linear inter polation 768
4. Numerical Example 770
Acknowledgments 772
REFERENCES 772
CHAPTER 27. TWO TYPES OF PIECEWISE QUADRATIC SPACES AND THEIR ORDER OF ACCURACY FOR POISSON'S EQUATION 774
REFERENCES 778
CHAPTER 28. A METHOD OF GALERKIN TYPE ACHIEVING OPTIMUM L2 ACCURACY FOR FIRST ORDER HYPERBOLICS AND EQUATIONS OF SCHRODINGER TYPE 780
REFERENCES 782
CHAPTER 29. RICHARDSON EXTRAPOLATION FOR PARABOLIC GALERKIN METHODS 784
CHAPTER 30. GEOMETRIC ASPECTS OF THE FINITE ELEMENT METHOD 786
1· Summary 786
2. Introduction 787
3. Univalent Mappings and Curvilinear Coordinate Systems. 789
4. Applications to boundary value problems. 794
Acknowl edgpents 798
REFERENCES 798
CHAPTER 31. THE USE OF INTERPOLATORY POLYNOMIALS FOR A FINITE ELEMENT SOLUTION OF THE MULTIGROUP DIFFUSION EQUATION 802
REFERENCES 804
CHAPTER 32. A "LOCAL'' BASIS OF GENERALIZED SPLINES OVER RIGHT TRIANGLES DETERMINED FROM A NONUNIFORM PARTITIONING OF THE PLANE 808
CHAPTER 33. LEAST SQUARE POLYNOMIAL SPLINE APPROXIMATION 810
REFERENCE 813
CHAPTER 34. SUBSPACES WITH ACCURATELY INTERPOLATED BOUNDARY CONDITIONS 814