Splines and Variational Methods Book

Chapter 1 Introductory Ideas 1

1.1 A Simply Stated Problem 1

1.2 Linear Spaces 2

1.3 Normed Linear Spaces 6

1.4 The Space L[subscript 2 a, b] 11

1.5 Basis for a Linear Space 15

1.6 Approximating from Finite Dimensional Subspaces 19

Chapter 2 Lagrangian Interpolates 24

2.1 Introduction 24

2.2 On Polynomials 26

2.3 Lagrange Interpolation 29

2.4 Computation and Choice of Basis 32

2.5 Error Estimates for Lagrange Interpolates 35

2.6 Best Approximation and Extended Error Estimates 40

2.7 Piecewise Lagrange Interpolation 44

Chapter 3 Hermitian Interpolates 52

3.1 Introduction 52

3.2 Computation of Piecewise Cubic Hermites 56

3.3 A Simple Application 60

3.4 Hermite Interpolation 63

3.5 Piecewise Hermite Interpolation 68

3.6 Computation of Piecewise Hermite Polynomials 70

3.7 The Hermite-Birkhoff Interpolation Problem 74

Chapter 4 Polynomial Splines and Generalizations 77

4.1 Introduction 77

4.2 Cubic Splines 78

4.3 Derivation of the B Splines 87

4.4 Splines and Ordinary Differential Equations 94

4.5 Error Analysis 107

Chapter 5 Approximating Functions of Several Variables 116

5.1 Surface Fitting 116

5.2 Approximates on a Rectangular Grid 118

5.3 Tensor Products 135

5.4 Approximates on Triangular Grids 137

5.5 Automatic Mesh Generation and Isoparametric Transforms 155

5.6 Blended Interpolates and Surface Approximation 168

Chapter 6 Fundamentals for Variational Methods 174

6.1 Variational Methods 174

6.2 Linear Operators 177

6.3 Inner Product Spaces 182

6.4 Norms, Convergence, and Completeness 187

6.5 Equivalent Norms 190

6.6 Best Approximations 192

6.7 Least Squares Fits 197

Chapter 7 The Finite Element Method201

7.1 Introduction 201

7.2 A Simple Application 205

7.3 An Elementary Error Analysis 211

7.4 Lowering the Smoothness Requirements-Choice of Linear Space 217

7.5 Some Practical Considerations 225

7.6 Applications to the Dirichlet Problem 227

7.7 The Mixed Boundary Value Problem 240

7.8 The Neumann Problem 245

7.9 Coerciveness and Rates of Convergence 251

7.10 Curved Boundaries and Nonconforming Elements 255

7.11 Higher Order Linear Ordinary Differential Equations 257

7.12 Second and Higher Order Elliptic Partial Differential Equations 262

7.13 Galerkin Methods and Least Squares Methods 267

Chapter 8 The Method of Collocation 273

8.1 Introduction 273

8.2 A Simple Special Case: Existence Via Matrix Analysis 279

8.3 Green's Functions 286

8.4 Collocation Existence Via Green's Functions 289

8.5 Error Analyses Via Green's Functions 296

8.6 Collocation and Partial Differential Equations 298

8.7 Orthogonal Collocation 304

8.8 A Connection Between Collocation and Galerkin Methods 314

Glossary of Symbols 319

Index 321