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AuthorAtkinson, Kendall. author
TitleTheoretical Numerical Analysis [electronic resource] : A Functional Analysis Framework / by Kendall Atkinson, Weimin Han
ImprintNew York, NY : Springer New York, 2001
Connect tohttp://dx.doi.org/10.1007/978-0-387-21526-6
Descript XVI, 451 p. 26 illus. online resource

SUMMARY

Overall, the book is clearly written, quite pleasant to read, and contains a lot of important material; and the authors have done an excellent job at balancing theoretical developments, interesting examples and exercises, numerical experiments, and bibliographical references. - R. Glowinski, SIAM Review


CONTENT

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 Newtonโ{128}{153}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 -- 8.2 The 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 Nystrรถm 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


Mathematics Mathematical analysis Analysis (Mathematics) Numerical analysis Mathematics Analysis Numerical Analysis



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