Analysis of approximation methods for differential and integral equations.

*(English)*Zbl 0574.41001
Applied Mathematical Sciences, 57. New York etc.: Springer-Verlag. XI, 398 p. DM 138.00 (1985).

This book treats numerous methods developed for obtaining approximate solutions to three classes of practically important problems, namely: Boundary-value problems in ordinary and partial differential equations (elliptic kind); Initial-value problems in partial differential equations and, finally, integral equations of the second kind. The methods considered include finite-difference methods, projection methods (Ritz and variants of Galerkin) and quadrature methods. The aim of the author is to present a unifying convergence theory through which one can prove the convergence of specific numerical methods with accompanying error estimates.

The book consists of thirteen chapters divided into four parts. Part I is devoted to the presentation of the approximation methods as well as the results concerning the solvability of the underlying approximate equations. The material in this part is useful to acquaint the reader with known schemes for approximating problems. In part II a unified theory of convergence is developed by relying upon the concept of ”discrete convergence”. The results of this part are applied to the problems cited above in parts III and IV.

The convergence theory given in part II is developed in a very general setting but is restricted to problems where the approximating equations are expressed in terms of equicontinuously equidifferentiable mapping. So, it is applicable to a series of classes of both linear and nonlinear problems and permits one to obtain two-sided error estimates. At some places comments are made concerning extensions of the results. The level of the book is suitable for advanced students and researchers who have a basic knowledge of numerical analysis and functional analysis. It may also serve as a reference for a series of well-known and other numerical schemes for the problem classes cited above.

The book consists of thirteen chapters divided into four parts. Part I is devoted to the presentation of the approximation methods as well as the results concerning the solvability of the underlying approximate equations. The material in this part is useful to acquaint the reader with known schemes for approximating problems. In part II a unified theory of convergence is developed by relying upon the concept of ”discrete convergence”. The results of this part are applied to the problems cited above in parts III and IV.

The convergence theory given in part II is developed in a very general setting but is restricted to problems where the approximating equations are expressed in terms of equicontinuously equidifferentiable mapping. So, it is applicable to a series of classes of both linear and nonlinear problems and permits one to obtain two-sided error estimates. At some places comments are made concerning extensions of the results. The level of the book is suitable for advanced students and researchers who have a basic knowledge of numerical analysis and functional analysis. It may also serve as a reference for a series of well-known and other numerical schemes for the problem classes cited above.

Reviewer: M.Idemen

##### MSC:

41-02 | Research exposition (monographs, survey articles) pertaining to approximations and expansions |

34A45 | Theoretical approximation of solutions to ordinary differential equations |

35A35 | Theoretical approximation in context of PDEs |

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |

45L05 | Theoretical approximation of solutions to integral equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |