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This volume provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear variational problems. The author avoids giving the results in the most general, abstract form so that it is easier for the reader to understand more clearly the essential ideas involved. Many examples are included to show the usefulness of the derived error estimates. Audience This volume is suitable for researchers and graduate students in applied and computational mathematics, and in engineering.

Error analysis (Mathematics) --- Duality theory (Mathematics) --- Algebra --- Mathematical analysis --- Topology --- Errors, Theory of --- Instrumental variables (Statistics) --- Mathematical statistics --- Numerical analysis --- Statistics

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Numerical analysis --- numerieke analyse

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These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.

Engineering & Applied Sciences --- Civil & Environmental Engineering --- Operations Research --- Applied Mathematics --- Spherical harmonics. --- Spherical functions. --- Functions, Spherical --- Functions, Potential --- Potential functions --- Mathematics. --- Approximation theory. --- Integral equations. --- Partial differential equations. --- Special functions. --- Numerical analysis. --- Physics. --- Numerical Analysis. --- Special Functions. --- Approximations and Expansions. --- Integral Equations. --- Partial Differential Equations. --- Physics, general. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Mathematical analysis --- Special functions --- Partial differential equations --- Equations, Integral --- Functional equations --- Functional analysis --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Harmonic analysis --- Harmonic functions --- Functions, special. --- Differential equations, partial.

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This textbook prepares graduate students for research in numerical analysis/computational mathematics by giving to them a mathematical framework embedded in functional analysis and focused on numerical analysis. This helps the student to move rapidly into a research program. The text covers basic results of functional analysis, approximation theory, Fourier analysis and wavelets, iteration methods for nonlinear equations, finite difference methods, Sobolev spaces and weak formulations of boundary value problems, finite element methods, elliptic variational inequalities and their numerical solution, numerical methods for solving integral equations of the second kind, and boundary integral equations for planar regions. The presentation of each topic is meant to be an introduction with certain degree of depth. Comprehensive references on a particular topic are listed at the end of each chapter for further reading and study. In this new edition many sections from the first edition have been revised to varying degrees as well as over 140 new exercises added. A new chapter on Fourier Analysis and wavelets has been included. Review of earlier edition: "...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, 2003.

Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Numerical analysis. --- Analysis. --- Numerical Analysis. --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis --- Functional analysis. --- 517.1 Mathematical analysis --- Calculus. --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Functions --- Geometry, Infinitesimal

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Mathematical analysis --- Numerical analysis --- analyse (wiskunde) --- numerieke analyse