Series Preface
Preface
1 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
2 Linear Operators on Normed Spaces
2.1 Operators
2.2 Continuous linear operators
2.2.1 C(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 C(D)
2.8.2 Properties of compact operators
2.8.3 Integral operators on L2(a,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 R(r) as a holomorphic function
3 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 or 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
4 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
5 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 Ggteaux 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
6 Finite Difference Method
6.1 Finite difference approximations
6.2 Lax equivalence theorem
6.3 More on convergence
7 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 Approximations
11.1 From variational equations to variational inequalities
11.2 Existence and uniqueness based on convex minimization
11.3 Existence and uniqueness results for a family of EVIs
11.4 Numerical approximations
11.5 Some contact problems in elasticity
11.5.1 A frictional contact problem
11.5.2 A Signorini frictionless contact problem
12 Numerical Solution of Fredholm Integral Equations of the Second Kind
12.1 Projection methods: General theory
12.1.1 Collocation methods
12.1.2 Galerkin methods
12.1.3 A general theoretical framework
12.2 Examples
12.2.1 Piecewise linear collocation
12.2.2 Trigonometric polynomial collocation
12.2.3 A piecewise linear Galerkin method
12.2.4 A Galerkin method with trigonometric polynomials
12.3 Iterated projection methods
12.3.1 The iterated Galerkin method
12.3.2 The iterated collocation solution
12.4 The Nystrom method
12.4.1 The NystrSm method for continuous kernel functions
12.4.2 Properties and error analysis of the NystrSm method
12.4.3 Collectively compact operator approximations
12.5 Product integration
12.5.1 Error analysis
12.5.2 Generalizations to other kernel functions
12.5.3 Improved error results for special kernels
12.5.4 Product integration with graded meshes
12.5.5 The relationship of product integration and collocation methods
12.6 Iteration methods
12.6.1 A two-grid iteration method for the Nystrom method
12.6.2 Convergence analysis
12.6.3 The iteration method for the linear system
12.6.4 An operations count
12.7 Projection methods for nonlinear equations
12.7.1 Linearization
12.7.2 A homotopy argument
12.7.3 The approximating finite-dimensional problem
13 Boundary Integral Equations
13.1 Boundary integral equations
13.1.1 Green's identities and representation formula
13.1.2 The Kelvin transformation and exterior problems
13.1.3 Boundary integral equations of direct type
13.2 Boundary integral equations of the second kind
13.2.1 Evaluation of the double layer potential
13.2.2 The exterior Neumann problem
13.3 A boundary integral equation of the first kind
13.3.1 A numerical method
14 Multivariable Polynomial Approximations
14.1 Notation and best approximation results
14.2 0rthogonal polynomials
14.2.1 Triple recursion relation
14.2.2 The orthogonal projection operator and its error
14.3 Hyperinterpolation
14.3.1 The norm of the hyperinterpolation operator
14.4 A Galerkin method for elliptic equations
14.4.1 The Galerkin method and its convergence
References
Index