Splines and Variational Methods

by P. M. Prenter

Overview

One of the clearest available introductions to variational methods, this text requires only a minimal background in calculus and linear algebra. Its self-contained treatment explains the application of theoretic notions to the kinds of physical problems that engineers regularly encounter. The text's first half concerns approximation theoretic notions, exploring the theory and computation of one- and two-dimensional polynomial and other spline functions. Later chapters examine variational methods in the solution of operator equations, focusing on boundary value problems in one and two dimensions. Additional topics include least squares and other Galerkin methods. Many helpful definitions, examples, and exercises appear throughout the book. A classic reference in spline theory, this volume will benefit experts as well as students of engineering and mathematics.

Product Details

ISBN-13:	9780486469027
Publisher:	Dover Publications
Publication date:	12/18/2008
Series:	Dover Books on Mathematics Series
Pages:	336
Product dimensions:	6.10(w) x 9.10(h) x 0.90(d)

Table of Contents

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