An introduction to the mathematical theory of finite elements

목차 (Table of Contents)

• CONTENTS
• Chapter 1 Introduction = 1 1
• 1.1 The Finite-Element Method = 1
• 1.2 The Mathematics of Finite Elements = 2
• 1.3 The Present Study = 3
• 1.4 Notations and Preliminaries = 4
• References = 5
• PART Ⅰ MATHEMATICAL FOUNDATIONS
• Chapter 2 Distributions, Mollifiers, and Mean Functions = 9
• 2.1 Introduction = 9
• 2.2 Functionals and Test Functions on One-Dimensional Domains = 10
• 2.3 Distributions = 14
• 2.4 Locally Integrable Generators, Regular and Singular Distributions = 17
• 2.5 Some Properties of Distributions = 20
• 2.6 Distributional Differential Equations = 26
• 2.7 Distributions and Generalized Functions in Rⁿ = 31
• 2.8 Fourier Transforms, Rapidly Decaying Functions, and Tempered Distributions = 36
• 2.9 Weak and Strong Derivatives in $$L_P$$(Ω) = 45
• 2.10 Mollifiers and Mean Functions = 46
• References = 54
• Chapter 3 Theory of Sobolev Spaces = 55
• 3.1 Introduction = 55
• 3.2 The Sobolev Space $$W_P^m$$(Ω) = 55
• 3.3 Partitions of Unity, Boundaries, and Cone Conditions = 57
• 3.4 Some Properties of the Sobolev Spaces $$W_P^m$$(Ω) and $$\dot W_P^m$$(Ω) = 61
• 3.5 The Sobolev Integral Identity = 67
• 3.6 The Sobolev Embedding Theorems = 79
• 3.7 The Decomposition of $$W_P^m$$(Ω) = 82
• References = 88
• Chapter 4 Hilbert Space Theory of Traces and Intermediate Spaces = 89
• 4.1 Introduction = 89
• 4.2 Hilbert Spaces $$H^m$$(Ω) of Integer Order = 90
• 4.3 Hilbert Spaces $$H^s$$ ($$R^n$$) for Real s ≥ 0 = 92
• 4.4 Duals of Hilbert Spaces = 96
• 4.5 Duals of Spaces $$H^s$$ ($$R^n$$)and $$H^m$$(Ω) = 104
• 4.6 The Trace Theorem for $$H^m$$($$R_+^n$$) = 112
• 4.7 Intermediate and Interpolation Spaces = 121
• 4.8 Interpolation Theory in Hilbert Spaces = 128
• 4.9 Hilbert Spaces $$H^s$$($$\partial $$Ω) = 137
• 4.10 The Trace Theorem for $$H^s$$(Ω) = 141
• References = 143
• Chapter 5 Some Elements of Elliptic Theory = 145
• 5.1 Introduction = 145
• 5.2 Linear Elliptic Operators = 146
• 5.3 Boundary Conditions = 152
• 5.4 Green's Formulas = 162
• 5.5 Regularity Theory in $$H^s$$(Ω), s ≥ 2m = 169
• 5.6 Compatibility Conditions - Existence and Uniqueness in $$H^s$$(Ω), s ≥2m = 176
• 5.7 Existence and Regularity Theory in $$H^s$$(Ω), s<2m = 182
• References = 192
• PART Ⅱ THE THEORY OF FINITE ELEMENTS
• Chapter 6 Finite-Element Interpolation = 197
• 6.1 Introduction = 197
• 6.2 Connectivity of Finite-Element Models of Domains Ω⊂ $$R^n$$= 198
• 6.3 Local and Global Representations of Functions = 206
• 6.4 Restrictions, Prolongations, and Projections = 215
• 6.5 Conjugate Basis Functions = 221
• 6.6 Finite-Element Families = 235
• 6.7 Accuracy of Finite-Element Interpolations = 264
• References = 283
• Chapter 7 Variational Boundary-Value Problems = 286
• 7.1 Introduction = 286
• 7.2 Formulation of Variational Boundary-Value Problems = 289
• 7.3 Coercive Bilinear Forms = 300
• 7.4 Weak Coerciveness = 310
• 7.5 Existence and Uniqueness of Solutions = 315
• References = 321
• Chapter 8 Finite-Element Approximations of Elliptic Boundary-Value Problems = 323
• 8.1 Introduction = 323
• 8.2 Galerkin Approximations = 323
• 8.3 Existence and Uniqueness of Galerkin Approximations = 326
• 8.4 Finite-Element Approximations = 330
• 8.5 Properties of Finite-Element Subspaces = 334
• 8.6 Error Estimates = 342
• 8.7 Pointwise and $$L_\infty $$(Ω) Error Estimates = 348
• 8.8 Quadrature, Boundary, and Data Errors = 350
• 8.9 $$H^{-1}$$ Finite-Element Methods = 365
• 8.10 Hybrid and Mixed Finite-Element Methods = 368
• References = 387
• Chapter 9 Time-Dependent Problems = 390
• 9.1 Introduction = 390
• 9.2 Finite-Element Models of the Diffusion Equation = 391
• 9.3 Semidiscrete L₂ Galerkin Approximations = 393
• 9.4 Elements of Semigroup Theory = 395
• 9.5 Semigroup Methods for Galerkin Approximations = 401
• 9.6 Hyperbolic Equations of Second Order = 409
• 9.7 First-Order Hyperbolic Equations = 415
• References = 418
• Author Index = 421
• Subject Index = 423