Analysis of Approximation Methods for Differential and Integral Equations
This book is primarily based on the research done by the Numerical Analysis Group at the Goethe-Universitat in Frankfurt/Main, and on material presented in several graduate courses by the author between 1977 and 1981. It is hoped that the text will be useful for graduate students and for scientists interested in studying a fundamental theoretical analysis of numerical methods along with its application to the most diverse classes of differential and integral equations. The text treats numerous methods for approximating solutions of three classes of problems: (elliptic) boundary-value problems, (hyperbolic and parabolic) initial value problems in partial differential equations, and integral equations of the second kind. The aim is to develop a unifying convergence theory, and thereby prove the convergence of, as well as provide error estimates for, the approximations generated by specific numerical methods. The schemes for numerically solving boundary-value problems are additionally divided into the two categories of finite difference methods and of projection methods for approximating their variational formulations.
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PRESENTATION OF NUMERICAL METHODS
Projection Methods for Variational Equations
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a-regularity analysis approximation methods arbitrary associated assume assumptions Banach spaces biconvergence bijective boundary-value problem C(Gn Chapter coefficients completely continuous consistency sequence continuously differentiable Crank-Nicolson method defined denote difference quotients differential equations Dirichlet boundary conditions discrete approximation discrete convergence discrete L norms discretely compact domain of definition equicontinuous equicontinuously equidifferentiable equivalent examples exists finite finite-difference approximations finite-difference methods following theorem Frechet-derivatives Galerkin methods given heat equation Hilbert space initial value problems integral equations integral operator inverse stability inverse stability inequality k+1 k+1 kernel Lax-Milgram Lemma Lax-Wendroff method Lemma linear mappings linear operator matrices maximum norms mesh widths moreover nonlinear obtain partial derivatives piecewise projection methods Proof Property 7.6 respect satisfied scalar Section semihomogeneous Stummel subset subspaces supremum norm surjective system of equations tion truncation errors uniform boundedness uniformly bounded uniquely solvable yields