11R1. Adaptive Meshing with Boundary Elements. Topics in Engineering, Vol 41. - JC Miranda-Valenzuela (ITESM Campus Toluca, Mexico) and KH Muci-Kuchler (Univ of Detroit Mercy, Detroit MI 48219). WIT Press, Southampton, UK. 2002. 293 pp. ISBN 1-85312-888-0. $169.00.

Reviewed by J Trevelyan (Sch of Eng, Univ of Durham, Durham, DH1 3LE, UK).

There has been considerable literature over the last 20 years surrounding adaptive methods of mesh refinement, or solution improvement, in the finite element method. Publications relating to the use of these methods in the boundary element method (BEM) have followed. The field is now sufficiently mature for the production of a textbook that might summarize the developments, and that might provide a useful starting point for a BEM researcher who is hoping to make use of these techniques. This book serves the purpose well.

The book is not solely about adaptive methods. The first five chapters, indeed, make little reference to adaptivity, but instead present a background to the BEM upon which the later chapters can build in their presentation of the adaptive techniques. There are different levels on which the BEM can be described, ranging from the intuitive to the pure mathematical. This book takes the mathematical approach. It is not for the mathematically faint-hearted, especially as more than the usual amount of space has been devoted to details of the tangent derivative formulations in which the authors have worked and which form the basis for the adaptive error indicators described towards the end of the book. On this point, though, the authors must have recognized that the market for the book is a relatively small set of researchers who have the mathematical skills already to prepare a BEM code, and who probably have prepared codes of their own already in which adaptivity might be implemented. It is because of the limited market, also, that I suspect the publishers have set a quite challenging price of $169 for a work running to 287 pages.

The book is generally well written and very readable. Many chapters include introductory sections containing very concise and well-presented literature reviews that read like a chronological story of developments. These sections provide a useful perspective to the field.

Chapter 6 is particularly useful, in which the authors present an overview of adaptive meshing and error estimation. The chapter would be a useful resource for a graduate student taking his/her first steps in preparation for implementing some adaptivity in a BEM code. The accumulation of ideas is presented in a logical, clear, and accessible way.

This reviewer has two criticisms: firstly, not enough use is made of examples. Those that are presented are generally clear and serve their purpose of illustrating the methods, but more examples would have aided clarity in places. Most examples illustrate the methods based around the Hermite elements favored by the authors, whereas more examples of conventional Lagrangian elements would have made the book a more general work. Secondly, the book has focused very heavily on the error estimation. Without question, error estimation is a key constituent of a successful adaptive scheme, but little space is given to the remeshing strategies that might be employed. In particular, little attention is given to p-adaptive schemes, or hp combinations.

In spite of these criticisms, and its cost this reviewer would recommend Adaptive Meshing with Boundary Elements as an introduction to adaptive methods for the BEM, and one that is justified now that these techniques are maturing.