Reliable methods for computer simulation. Error control and a posteriori estimates. (English) Zbl 1076.65093

Neittaanmäki, Pekka; Repin, Sergey

Studies in Mathematics and its Applications 33. Amsterdam: Elsevier (ISBN 0-444-51376-0/hbk). x, 305 p. (2004).

During the past decade, the a posteriori error analysis of numerical methods for partial differential equations has gained a lot of attention as documented by a series of monographs on this subject. There are various techniques including residual and hierarchical type estimators, those which are based on local averaging, and the so-called goal oriented weigthed dual method. The monograph under consideration contains a comprehensive treatment of yet another approach: duality error majorants. These are functional type a posteriori error estimators that result in computable error bounds for all types of conforming approximations.
The monograph provides all necessary prerequisites from the calculus of variations and concentrates on linear and some nonlinear variational problems including second and fourth order elliptic boundary value problems in the linear regime and stationary variational inequalities in the nonlinear case. The book is very well written and technically sound. It should not be missed on the bookshelves of those interested in modern methods for the numerical solution of partial differential equations.

Reviewer: Ronald H. W. Hoppe (Augsburg)

MSC:

65N15	Error bounds for boundary value problems involving PDEs
49J20	Existence theories for optimal control problems involving partial differential equations
49J40	Variational inequalities
35J25	Boundary value problems for second-order elliptic equations
35J40	Boundary value problems for higher-order elliptic equations
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis

Keywords:

a posteriori error analysis; duality error majorants; linear elliptic problems; textbook; computable error bounds; calculus of variations; nonlinear variational problems; stationary variational inequalities