Recent decades have seen a very rapid success in developing numerical methods based on explicit control over approximation errors. It may be said that nowadays a new direction is forming in numerical analysis, the main goal of which is to develop methods ofreliable computations. In general, a reliable numerical method must solve two basic problems: (a) generate a sequence of approximations that converges to a solution and (b) verify the accuracy of these approximations. A computer code for such a method must consist of two respective blocks: solver and checker. In this book, we are chiefly concerned with the problem (b) and try to present the main approaches developed for a posteriori error estimation in various problems. The authors try to retain a rigorous mathematical style, however, proofs are constructive whenever possible and additional mathematical knowledge is presented when necessary. The book contains a number of new mathematical results and lists a posteriori error estimation methods that have been developed in the very recent time.

Key Features

· computable bounds of approximation errors · checking algorithms · iteration processes · finite element methods · elliptic type problems · nonlinear variational problems · variational inequalities


Mathematical modellers, civil engineers, computer scientists, mathematical economists and mathematical biologists.

Table of Contents

Contents 1. Introduction. 2. Mathematical background. 3. A posteriori estimates for iteration methods. 4. A posteriori estimates for finite element approximations. 5. Foundations of duality theory. 6. Two-sided a posteriori estimates for linear elliptic problems. 7. A posteriori estimates for nonlinear variational problems. 8. A posteriori estimates for variational inequalities. Bibliography. Notation. Index.


No. of pages:
© 2004
Elsevier Science
eBook ISBN:
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About the authors

Pekka Neittaanmäki

Affiliations and Expertise

University of Jyväskylä, Finland

Sergey Repin

Affiliations and Expertise

V.A. Steklov Institute of Mathematics, St. Petersburg, Russian Federation