Numerical Analysis, Softcover reprint of the original 1st ed. 1973

This book is an introduction to one of the important as­ pects of Numerical Analysis, namely the approximate solution of functional equations. We intend to show, by a few brief examples, the different theoretical and practical problems related to the numerical approximation of boundary value problems. We have chosen for this the approximate solution of certain linear elliptic partial differential equations (the first two parts of the book) and the approximate solution of a nonlinear elliptic differential equation. This book is not a systematic study of the subject, but the methods developed here can be applied to large classes of linear and nonlinear elliptic problems. The book assumes that the reader's knowledge of Anal­ ysis is comparable to what is taught in the first years of graduate studies. This means a good knowledge of Hilbert spaces, elements of measure theory and theory of distributions. The subject matter of the book covers the usual content of a first course on Numerical Analysis of partial differential equations.

Approximate Solution of Functional Equations: General Remarks.- I.: Approximate Solution of Linear Elliptic Equations.- 1. The Projection Theorem.- 2. The Method of Galerkin.- 3. The Approximation of Normed Spaces.- 4. Approximation of Linear Variational Problems.- 5. A Method for Computation of the Error.- 6. The Method of Fractionary Steps.- II: Applications.- 7. Spaces of Functions Associated with an Open Set in Rn.- 8. Approximation of Some Function Spaces by Finite Differences (I).- 9. Approximation of Some Function Spaces by Finite Differences (II).- 10. Approximation of Some Function Spaces by Finite Element Methods.- 11. Example I: The Dirichlet Problem.- 12. Example II: The Neumann Problem.- III: A Nonlinear Elliptic Equation.- 13. The Exact Problem.- 14. Approximate Problems.- Appendix: Topological Vector Spaces.