Accuracy Verification Methods_ Theory and Algorithms [Mali, Neittaanmaki & Repin 2013-10-27].pdf

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Accuracy Verification Methods
Computational Methods in Applied Sciences
Volume 32
Series Editor
E. Oñate
International Center for Numerical Methods in Engineering (CIMNE)
Technical University of Catalonia (UPC)
Edificio C-1, Campus Norte UPC
Gran Capitán, s/n
08034 Barcelona, Spain
onate@cimne.upc.edu
url:
http://www.cimne.com
For further volumes:
www.springer.com/series/6899
Olli Mali
r
Pekka Neittaanmäki
r
Sergey Repin
Accuracy
Verification
Methods
Theory and Algorithms
Olli Mali
Department of Mathematical Information
Technology
University of Jyväskylä
Jyväskylä, Finland
Pekka Neittaanmäki
Department of Mathematical Information
Technology
University of Jyväskylä
Jyväskylä, Finland
Sergey Repin
Steklov Institute of Mathematics
Russian Academy of Sciences
St. Petersburg, Russia
and
University of Jyväskylä
Jyväskylä, Finland
ISSN 1871-3033 Computational Methods in Applied Sciences
ISBN 978-94-007-7580-0
ISBN 978-94-007-7581-7 (eBook)
DOI 10.1007/978-94-007-7581-7
Springer Dordrecht Heidelberg New York London
Library of Congress Control Number: 2013951642
© Springer Science+Business Media Dordrecht 2014
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Preface
Nowadays, mathematical and numerical modeling has become an essential compo-
nent of the general scientific process. Ever since the 1960s, numerical analysis and
scientific computation have made up the most rapidly growing part of mathemat-
ics. One of the challenging problems in this area is the creation of fully reliable
computer simulation methods, which could become an adequate complement to ex-
perimental sciences. This book aims to give an overview of mathematical methods
and computer technologies focused on reliable verification of computed solutions
and present recently developed methods. We hope that it will be useful for an audi-
ence much larger than just advanced specialists in numerical analysis and computer
simulation methods. In actuality, the book can be used in three different ways.
For engineers and specialists in natural sciences interested in quantitative analy-
sis of mathematical models, it is best to concentrate on algorithms and prescriptions,
which explain how to measure the accuracy of a numerical solution. In Chap.
2,
we
discuss various error indicators, which are used in mesh adaptive numerical algo-
rithms in order to achieve proper restructuring (refinement) of the computational
mesh (or changing the set of trial functions). We suggest a unified approach to this
question and discuss different error indicators. Chapter
3
is concerned with the ques-
tion: “how can guaranteed and computable bounds of errors associated with approx-
imations of differential equations be derived?”. We tried to explain this in simple
terms without a deep excursion into the mathematical background. In other words,
the reader whose main purpose is to use the results (estimates) will find the corre-
sponding detailed recommendations. Certainly, they are given for a limited amount
of typical problems. Other cases can be found in the literature cited or require addi-
tional analysis (in the latter case, a good understanding of the mathematical theory
is necessary).
For advanced specialists interested in the development of new error estimation
methods, Chaps.
3–5
are the most interesting. Here, we discuss mathematical tech-
nologies that provide guaranteed error control and applications to analysis of prob-
lems with uncertain data. These chapters essentially use materials exposed in the
books P. Neittaanmäki and S. Repin [NR04] and S. Repin [Rep08] (in [NR04] the
reader can find a complete set of a posteriori error estimation theory generated by
v

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