Today’s problems in scientific computing have become so large that they frequently cannot be handled in an acceptable amount of time on a single-processor system. Thus, there is a substantial demand for efficient parallel algorithms. In the common case of problems that can be mathematically modeled as partial differential equations, a frequently used standard method is based on domain decomposition techniques. This book presents a collection of 13 chapters devoted to various aspects of such techniques. Each of the chapters is written by a small group of specialists in a particular topic. The chapters are written independently of each other; the authors only infrequently elaborate on the interconnections between their own topic and the topics of the other chapters.

The first three chapters deal with the problem of finding a suitable decomposition of the basic domain of the differential equation and the closely related problem of partitioning graphs. Specifically, chapter 1 gives an overview of the currently available methods and software packages and an outlook of some promising approaches for future extensions. Chapter 2 then provides a detailed description of JOSTLE, a particular software package for graph partitioning. Finally, in chapter 3, the reader can find a thorough review of an important auxiliary tool in this context, namely the VTK library that can be used for the visualization of partitioned graphs or decomposed meshes. These three chapters form a nice introduction for beginners who want to get started in the field.

The remaining ten chapters are devoted to the study of the mathematical aspects of the domain decomposition principle. The main questions here are: What effects does such an approach have on the equation systems that need to be solved? How can we find the properly modified equations? What are their properties? How can we solve them? Of course, there any many possible answers to these questions, corresponding to the various algorithms that have been proposed in the past. Therefore, chapters 4 and 5 give an overview of the general basic concepts and problems from the point of view of numerical mathematics, and establish the relationships to the aspects discussed in the first three chapters.

Chapters 6 to 13 are more specialized. One concrete algorithm, the FETI-DP scheme, is described in detail in chapter 6. This is followed by chapter 7 that deals with methods for a special class of problems, namely those that lead to nonsymmetric and indefinite systems of equations. The authors of chapter 8 elaborate on the relationships between domain decomposition and the preconditioning of systems of equations that need to be solved, using mainly an algebraic perspective. The focus of chapter 9 is on the development of an efficient method of the optimized Schwarz type for a system of equations arising in a very specific application related to the three-dimensional modeling of noise in an automobile exhaust system. Similarly, chapter 10 is devoted to special domain decomposition methods for the modeling of incompressible flows. Chapter 11 discusses the principle of extrapolation that can be used as a convergence acceleration technique in a very general setting. The combination of domain decomposition and the large time increment (LATIN) method is shown, in chapter 12, to be a useful approach for such problems that involve contact conditions. Finally, chapter 13 gives some information about a mortar method for the modeling of elastic plates.

A regrettable feature of the book is the keyword index at the end. It has a relatively small number of entries and, what is worse, in all but one case, the page numbers of the entries refer to the first page of the chapter where the keyword appears instead of the page where the keyword can actually be found. An experienced reader may be able to find his way to a particular piece of information in which he may be interested, but, for a novice, it’s a lot of unnecessary work.

In spite of this criticism, I think that the book presents a good survey of the state of the art in an important area of scientific computing. It is likely to be helpful to both novices and advanced readers who are looking for help in the implementation of domain decomposition techniques for their numerical problems.