\section*{Preface}%+
\addcontentsline{toc}{chapter}{Preface}
\markboth{PREFACE}{PREFACE}

%``What was it in the middle part?''
%  Kevin Kline in a Fish named Wanda.


During the last years, scientific computing has become an important
research branch located between applied mathematics and applied
sciences and engineering. Nowadays, in numerical mathematics not only
simple model problems are treated, but modern and well-founded
mathematical algorithms are applied to solve complex problems of real
life applications. Such applications are demanding for computational
realization and need suitable and robust tools for a flexible and
efficient implementation. Modularity and abstract concepts allow
for an easy transfer of methods to different applications.

Inspired by and parallel to the investigation of  real life applications,
numerical mathematics has built and improved many modern algorithms
which are now standard tools in scientific computing. Examples
are adaptive methods, higher order discretizations,
fast linear and non-linear iterative solvers, multi-level algorithms, etc.
These mathematical tools are able to reduce computing times
tremendously and for many applications a simulation can only be realized 
in a reasonable timeframe using such highly efficient algorithms.

A very flexible software is needed when working in both fields
of scientific computing and numerical mathematics. We developed the
toolbox \textsf{ALBERTA}\footnote{The original name of the toolbox was
  \textsf{ALBERT}. Due to copyright reasons, we had to rename it and
  we have chosen \ALBERTA.} for meeting these requirements.  Our
intention in the design of \ALBERTA is threefold: First, it is a
toolbox for fast and flexible implementation of efficient software for
real life applications, based on the modern algorithms mentioned
above.  Secondly, in an interplay with mathematical analysis, \ALBERTA
is an environment for improving existent, or developing new numerical
methods. And finally, it allows the direct integration of such new or
improved methods in existing simulation software.

Before having \ALBERTA, we worked with a variety of solvers, each designed for
the solution of one single application. Most of them were based on data
structures specifically designed for one single application. A combination of
different solvers or exchanging modules between programs was hard to do.
Facing these problems, we wanted to develop a general adaptive finite element
environment, open for implementing a large class of applications, where an
exchange of modules and a coupling of different solvers is easy to realize.

Such a toolbox has to be based on a solid concept which is still open
for extensions as science develops. Such a solid concept can be
derived from a mathematical abstraction of problem classes, 
numerical methods, and solvers. Our mathematical view of
numerical algorithms, especially finite element methods, is based on
our education and scientific research in the departments for applied
mathematics at the universities of Bonn and Freiburg. This view point
has greatly inspired the abstract concepts of \ALBERTA as well as
their practical realization, reflected in the main data structures.
The robustness and flexible extensibility of our concept was approved
in various applications from physics and engineering, like
computational fluid dynamics, structural mechanics, industrial crystal
growth, etc.\ as well as by the validation of new mathematical methods.

\ALBERTA is a library with data structures and functions for adaptive
finite element simulations in one, two, and three space dimension,
written in the programming language ANSI-C. Shortly after finishing the
implementation of the first version of \ALBERTA and using it for first
scientific applications, we confronted students with it in a course
about finite element methods. The idea was to work on more interesting
projects in the course and providing a strong foundation for an
upcoming diploma thesis.  Using \ALBERTA in education then required a
documentation of data structures and functions. The numerical course
tutorials were the basis for a description of the background and
concepts of adaptive finite elements.

The combination of the abstract and concrete description resulted in a
manual for \ALBERTA and made it possible that it is now used world
wide in universities and research centers.  The interest from other
scientists motivated a further polishing of the manual as well as the
toolbox itself, and resulted in this book, which is accompanied by a full
distribution of \ALBERTA on a CD. 

\bigskip

Starting first as a two-men-project, \ALBERTA is evolving and now
there are more people maintaining and extending it. We are
grateful for a lot of substantial contributions coming from: Michael
Fried, who was the first brave man besides us to use
\textsf{ALBERT}, Claus-Justus Heine, Daniel K\"oster, and Oliver
Kriessl. Daniel and Claus in particular set up the GNU configure tools
for an easy, platform-independent installation of the software.

We are indebted to the authors of the gltools, especially J\"urgen
Fuhrmann, and also to the developers of GRAPE, especially Bernard
Haasdonk, Robert Kl\"ofkorn, Mario Ohlberger, and Martin Rumpf.

We want to thank the Department of Mathematics at the University of
Maryland (USA), in particular Ricardo H. Nochetto, where part of the
documentation was written during a visit of the second author.  We
appreciate the invitation of the Isaac Newton Institute in Cambridge
(UK) where we could meet and work intensively on the revision of the
manual for three weeks.

We thank our friends, distributed all over the world, who have pointed
out a lot of typos in the manual and suggested several
improvements for \ALBERTA.

Last but not least, \ALBERTA would not have come into being
without the stimulating atmosphere in the group in Freiburg, which was
the perfect environment for working on this project. We want to express
our gratitude to all former colleagues, especially Gerhard Dziuk.

\bigskip\bigskip

\rightline{Bremen and Augsburg, December 2003}
\medskip

\rightline{Alfred Schmidt and Kunibert G. Siebert}
\vfill


The book is organized as follows: In Chapter 1 we describe the
concepts of adaptive finite element methods and its ingredients like
the domain discretization, finite element basis functions and degrees
of freedom, numerical integration via quadrature formulas for the
assemblage of discrete systems, and adaptive algorithms.

The second chapter is a tutorial for using \ALBERTA without giving
much details about data structures and functions. The implementation
of three model problems is presented and explained.  We start with the
easy and straight forward implementation of the Poisson problem to
learn about the basics of \ALBERTA\@. The examples with the
implementation of a nonlinear reaction-diffusion problem and the time
dependent heat equation are more involved and show the tools of
\ALBERTA for attacking more complex problems.  The chapter is closed
with a short introduction to the installation of the
\ALBERTA-distribution enclosed to this book in a UNIX/Linux
environment. 
%This distribution contains the source files for the
%example problems described here.

The realization of data structures and functions in \ALBERTA is
based on the abstract concepts presented in Chapter 1. A detailed
description of all data structures and functions of \ALBERTA is
given in Chapter 3. The book closes with separate lists of all data types,
symbolic constants, functions, and macros.

\bigskip

\textbf{Tentative:}
The cover pictures show the \ALBERTA logo, 
cogwheel \cite{BoehmSchmidtWolff:03},
flute \cite{BambergerBaenschSiebert:01}, 
and minimal surface \cite{DoerflerSiebert:03},
and perhaps also Bridgman crystal growth \cite{BBSBBDK:2003, LBDBKSSD:02}.

\bigskip

Visit the \ALBERTA web site, \url{www.alberta-fem.de}, for more
information, FAQ, contributions, updates, pictures from different projects,
etc.  

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "alberta"
%%% End: