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viii Preface to the English Edition

thank senior editor Achi Dosanjh, from Springer-Verlag New York, Inc., for
her constant encouragement.

Remarks for the Reader and the Use in Lectures
The size of the text corresponds roughly to four hours of lectures per week
over two terms. If the course lasts only one term, then a selection is nec-
essary, which should be orientated to the audience. We recommend the
following “cuts”:
Chapter 0 may be skipped if the partial differential equations treated
therein are familiar. Section 0.5 should be consulted because of the notation
collected there. The same is true for Chapter 1; possibly Section 1.4 may
be integrated into Chapter 3 if one wants to deal with Section 3.9 or with
Section 7.5.
Chapters 2 and 3 are the core of the book. The inductive presenta-
tion that we preferred for some theoretical aspects may be shortened for
students of mathematics. To the lecturer’s taste and depending on the
knowledge of the audience in numerical mathematics Section 2.5 may be
skipped. This might impede the treatment of the ILU preconditioning in
Section 5.3. Observe that in Sections 2.1–2.3 the treatment of the model
problem is merged with basic abstract statements. Skipping the treatment
of the model problem, in turn, requires an integration of these statements
into Chapter 3. In doing so Section 2.4 may be easily combined with Sec-
tion 3.5. In Chapter 3 the theoretical kernel consists of Sections 3.1, 3.2.1,
3.3–3.4.
Chapter 4 presents an overview of its subject, not a detailed development,
and is an extension of the classical subjects, as are Chapters 6 and 9 and
the related parts of Chapter 7.
In the extensive Chapter 5 one might focus on special subjects or just con-
sider Sections 5.2, 5.3 (and 5.4) in order to present at least one practically
relevant and modern iterative method.
Section 8.1 and the first part of Section 8.2 contain basic knowledge of
numerical mathematics and, depending on the audience, may be omitted.
The appendices are meant only for consultation and may complete
the basic lectures, such as in analysis, linear algebra, and advanced
mathematics for engineers.
Concerning related textbooks for supplementary use, to the best of our
knowledge there is none covering approximately the same topics. Quite a
few deal with finite element methods, and the closest one in spirit probably
is [21], but also [6] or [7] have a certain overlap, and also offer additional
material not covered here. From the books specialised in finite difference
methods, we mention [32] as an example. The (node-oriented) finite volume
method is popular in engineering, in particular in fluid dynamics, but to
the best of our knowledge there is no presentation similar to ours in a