Series Preface
Preface
Linear Spaces
1.1 Linear spaces
1.2 Normed spaces
1.2.1 Convergence
1.2.2 Banach spaces
1.2.3 Completion of normed spaces
1.3 Inner product spaces
1.3.1 Hilbert spaces
1.3.2 Orthogonality
1.4 Spaces of continuously differentiable functions
1.4.1 HSlder spaces
1.5 Lp spaces
1.6 Compact sets
Linear Operators on Normed Spaces
2.1 Operators
2.2 Continuous linear operators
2.2.1 V,W as a Banach space
2.3 The geometric series theorem and its variants
2.3.1 A generalization
2.3.2 A perturbation result
2.4 Some more results on linear operators
2.4.1 An extension theorem
2.4.2 Open mapping theorem
2.4.3 Principle of uniform boundedness
2.4.4 Convergence of numerical quadratures
2.5 Linear functionals
2.5.1 An extension theorem for linear functionals
2.5.2 The Riesz representation theorem
2.6 Adjoint operators
2.7 Weak convergence and weak compactness
2.8 Compact linear operators
2.8.1 Compact integral operators on CD
2.8.2 Properties of compact operators
2.8.3 Integral operators on L2a,b
2.8.4 The Fredholm alternative theorem
2.8.5 Additional results on Fredholm integral equations
2.9 The resolvent operator
2.9.1 RA as a holomorphic function
Approximation Theory
3.1 Approximation of continuous functions by polynomials
3.2 Interpolation theory
3.2.1 Lagrange polynomial interpolation
3.2.2 Hermite polynomial interpolation
3.2.3 Piecewise polynomial interpolation
3.2.4 Trigonometric interpolation
3.3 Best approximation
3.3.1 Convexity,lower semicontinuity
3.3.2 Some abstract existence results
3.3.3 Existence of best approximation
3.3.4 Uniqueness of best approximation
3.4 Best approximations in inner product spaces,projection on
closed convex sets
3.5 Orthogonal polynomials
3.6 Projection operators
3.7 Uniform error bounds
3.7.1 Uniform error bounds for L2-approximations
3.7.2 L2-approximations using polynomials
3.7.3 Interpolatory projections and their convergence
Fourier Analysis and Wavelets
4.1 Fourier series
4.2 Fourier transform
4.3 Discrete Fourier transform
4.4 Haar wavelets
4.5 Multiresolution analysis
Nonlinear Equations and Their Solution by Iteration
5.1 The Banach fixed-point theorem
5.2 Applications to iterative methods
5.2.1 Nonlinear algebraic equations
5.2.2 Linear algebraic systems
5.2.3 Linear and nonlinear integral equations
5.2.4 Ordinary differential equations in Banach spaces
5.3 Differential calculus for nonlinear operators
5.3.1 Frechet and Gateaux derivatives
5.3.2 Mean value theorems
5.3.3 Partial derivatives
5.3.4 The Gateaux derivative and convex minimization
5.4 Newton''s method
5.4.1 Newton''s method in Banach spaces
5.4.2 Applications
5.5 Completely continuous vector fields
5.5.1 The rotation of a completely continuous vector field
5.6 Conjugate gradient method for operator equations
Finite Difference Method
6.1 Finite difference approximations
6.2 Lax equivalence theorem
6.3 More on convergence
Sobolev Spaces
7.1 Weak derivatives
7.2 Sobolev spaces
7.2.1 Sobolev spaces of integer order
7.2.2 Sobolev spaces of real order
7.2.3 Sobolev spaces over boundaries
7.3 Properties
7.3.1 Approximation by smooth functions
7.3.2 Extensions
7.3.3 Sobolev embedding theorems
7.3.4 Traces
7.3.5 Equivalent norms
7.3.6 A Sobolev quotient space
7.4 Characterization of Sobolev spaces via the Fourier
transform
7.5 Periodic Sobolev spaces
7.5.1 The dual space
7.5.2 Embedding results
7.5.3 Approximation results
7.5.4 An illustrative example of an operator
7.5.5 Spherical polynomials and spherical harmonics
7.6 Integration by parts formulas
8 Weak Formulations of Elliptic Boundary Value Problems
8.1 A model boundary value problem
8.2 Some general results on existence and uniqueness
8.3 The Lax-Milgram Lemma
8.4 Weak formulations of linear elliptic boundary value
problems
8.4.1 Problems with homogeneous Dirichlet boundary
con-ditions
8.4.2 Problems with non-homogeneous Dirichlet boundary
conditions
8.4.3 Problems with Neumann boundary conditions
8.4.4 Problems with mixed boundary conditions
8.4.5 A general linear second-order elliptic boundary value
problem
8.5 A boundary value problem of linearized elasticity
8.6 Mixed and dual formulations
8.7 Generalized Lax-Milgram Lemma
8.8 A nonlinear problem
9 The Galerkin Method and Its Variants
9.1 The Galerkin method
9.2 The Petrov-Galerkin method
9.3 Generalized Galerkin method
9.4 Conjugate gradient method： variational formulation
10 Finite Element Analysis
10.1 One-dimensional examples
10.1.1 Linear elements for a second-order problem
10.1.2 High order elements and the condensation technique
10.1.3 Reference element technique
10.2 Basics of the finite element method
10.2.1 Continuous linear elements
10.2.2 Affine-equivalent finite elements
10.2.3 Finite element spaces
10.3 Error estimates of finite element interpolations
10.3.1 Local interpolations
10.3.2 Interpolation error estimates on the reference element
10.3.3 Local interpolation error estimates
10.3.4 Global interpolation error estimates
10.4 Convergence and error estimates
……
11 Elliptic Variational Inequalities and Their Numerical
Ap-proximations
12 Numerical Solution of Fredholm Integral Equations of the Second
Kind
13 Boundary Integral Equations
14 Multivariable Polynomial Approximations
References
Index
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