Numerical Treatment of Partial Differential Equations - Christian Grossmann.pdf

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Christian Grossmann
Hans-Görg Roos
Martin Stynes
Numerical Treatment
of Partial Differential
Equations
Universitext
Christian Grossmann
Hans-G¨ rg Roos
o
Numerical Treatment
of Partial Differential
Equations
Translated and revised by Martin Stynes
Prof. Dr. Christian Grossmann
Prof. Dr. Hans-G¨ rg Roos
o
Institute of Numerical Mathematics
Department of Mathematics
Technical University of Dresden
D-01062 Dresden, Germany
e-mail:
Christian.Grossmann@tu-dresden.de
Hans-Goerg.Roos@tu-dresden.de
Prof. Dr. Martin Stynes
School of Mathematical Sciences
Aras na Laoi
University College Cork
Cork, Ireland
e-mail:
m.stynes@ucc.ie
Mathematics Subject Classification (2000):
65N, 65F
Translation and revision of the 3rd edition of “Numerische Behandlung Partieller
Differentialgleichungen” Published by Teubner, 2005.
Library of Congress Control Number:
2007931595
ISBN:
978-3-540-71582-5
Springer Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the mater-
ial is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Dupli-
cation of this publication or parts thereof is permitted only under the provisions of the German
Copyright Law of September
9, 1965,
in its current version, and permission for use must always
be obtained from Springer. Violations are liable for prosecution under the German Copyright Law.
Springer is a part of Springer Science+Business Media
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Springer-Verlag Berlin Heidelberg
2007
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Typesetting by the authors and SPi using a Springer LTEX macro package
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Preface
Many well-known models in the natural sciences and engineering, and today
even in economics, depend on partial differential equations. Thus the efficient
numerical solution of such equations plays an ever-increasing role in state-of-
the-art technology. This demand and the computational power available from
current computer hardware have together stimulated the rapid development
of numerical methods for partial differential equations—a development that
encompasses convergence analyses and implementational aspects of software
packages.
In 1988 we started work on the first German edition of our book, which
appeared in 1992. Our aim was to give students a textbook that contained the
basic concepts and ideas behind most numerical methods for partial differen-
tial equations. The success of this first edition and the second edition in 1994
encouraged us, ten years later, to write an almost completely new version,
taking into account comments from colleagues and students and drawing on
the enormous progress made in the numerical analysis of partial differential
equations in recent times. The present English version slightly improves the
third German edition of 2005: we have corrected some minor errors and added
additional material and references.
Our main motivation is to give mathematics students and mathematically-
inclined engineers and scientists a textbook that contains all the basic discretiza-
tion techniques for the fundamental types of partial differential equations; one
in which the reader can find analytical tools, properties of discretization tech-
niques and advice on algorithmic aspects. Nevertheless, we acknowledge that
in fewer then 600 pages it is impossible to deal comprehensively with all these
topics, so we have made some subjective choices of material. Our book is
mainly concerned with finite element methods (Chapters 4 and 5), but we also
discuss finite difference methods (Chapter 2) and finite volume techniques.
Chapter 8 presents the basic tools needed to solve the discrete problems gener-
ated by numerical methods, while Chapter 6 (singularly perturbed problems)
and Chapter 7 (variational inequalities and optimal control) are special topics
that reflect the research interests of the authors.

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