Atkinson / Han Theoretical Numerical Analysis

A Functional Analysis Framework
Erscheinungsjahr 2006
ISBN: 978-0-387-21526-6
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 39, 451 Seiten, eBook

Reihe: Texts in Applied Mathematics

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this text book series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math ematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs.

Zielgruppe

Research

Autoren/Hrsg.

Atkinson, Kendall

Han, Weimin

Inhaltsverzeichnis

1 Linear Spaces.- 1.1 Linear spaces.- 1.2 Normed spaces.- 1.3 Inner product spaces.- 1.4 Spaces of continuously differentiable functions.- 1.5 Lp spaces.- 1.6 Compact sets.- 2 Linear Operators on Normed Spaces.- 2.1 Operators.- 2.2 Continuous linear operators.- 2.3 The geometric series theorem and its variants.- 2.4 Some more results on linear operators.- 2.5 Linear functional.- 2.6 Adjoint operators.- 2.7 Types of convergence.- 2.8 Compact linear operators.- 2.9 The resolvent operator.- 3 Approximation Theory.- 3.1 Interpolation theory.- 3.2 Best approximation.- 3.3 Best approximations in inner product spaces.- 3.4 Orthogonal polynomials.- 3.5 Projection operators.- 3.6 Uniform error bounds.- 4 Nonlinear Equations and Their Solution by Iteration.- 4.1 The Banach fixed-point theorem.- 4.2 Applications to iterative methods.- 4.3 Differential calculus for nonlinear operators.- 4.4 Newton’s method.- 4.5 Completely continuous vector fields.- 4.6 Conjugate gradient iteration.- 5 Finite Difference Method.- 5.1 Finite difference approximations.- 5.2 Lax equivalence theorem.- 5.3 More on convergence.- 6 Sobolev Spaces.- 6.1 Weak derivatives.- 6.2 Sobolev spaces.- 6.3 Properties.- 6.4 Characterization of Sobolev spaces via the Fourier transform.- 6.5 Periodic Sobolev spaces.- 6.6 Integration by parts formulas.- 7 Variational Formulations of Elliptic Boundary Value Problems.- 7.1 A model boundary value problem.- 7.2 Some general results on existence and uniqueness.- 7.3 The Lax-Milgram lemma.- 7.4 Weak formulations of linear elliptic boundary value problems.- 7.5 A boundary value problem of linearized elasticity.- 7.6 Mixed and dual formulations.- 7.7 Generalized Lax-Milgram lemma.- 7.8 A nonlinear problem.- 8 The Galerkin Method and Its Variants.- 8.1 The Galerkin method.- 8.2The Petrov-Galerkin method.- 8.3 Generalized Galerkin method.- 9 Finite Element Analysis.- 9.1 One-dimensional examples.- 9.2 Basics of the finite element method.- 9.3 Error estimates of finite element interpolations.- 9.4 Convergence and error estimates.- 10 Elliptic Variational Inequalities and Their Numerical Approximations.- 10.1 Introductory examples.- 10.2 Elliptic variational inequalities of the first kind.- 10.3 Approximation of EVIs of the first kind.- 10.4 Elliptic variational inequalities of the second kind.- 10.5 Approximation of EVIs of the second kind.- 11 Numerical Solution of Fredholm Integral Equations of the Second Kind.- 11.1 Projection methods: General theory.- 11.2 Examples.- 11.3 Iterated projection methods.- 11.4 The Nyström method.- 11.5 Product integration.- 11.6 Projection methods for nonlinear equations.- 12 Boundary Integral Equations.- 12.1 Boundary integral equations.- 12.2 Boundary integral equations of the second kind.- 12.3 A boundary integral equation of the first kind.- References.