الرئيسية

samedi 31 août 2019

Teoretical Numerical Analysis

Teoretical  Numerical Analysis

A Fonctinal Analysis Framework

Auteur : Kendall Atkinson • Weimin Han

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A Fonctinal Analysis Framework


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 H ̈older 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 L(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(λ) 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 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 

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 Fr ́echet and Gˆateaux derivatives  
5.3.2 Mean value theorems 
5.3.3 Partial derivatives 
5.3.4 The Gˆateaux 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 253
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 conditions 
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 Nystr ̈om method  
12.4.1 The Nystr ̈om method for continuous kernel functions 
12.4.2 Properties and error analysis of the Nystr ̈om 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 Nystr ̈om 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 Orthogonal 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  




Atkinson K., Han W. Theoretical numerical analysis. A functional analysis framework (3ed., Springer, 2009)(ISBN 1441904573)(O)(642s)_MN_.pdf - 3.3 MB

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