The finite-element method is a well-established mathematical tool that allows the use of computing techniques for the solution of problems arising in the sciences and engineering. As such, it is a topic that can be addressed from many different perspectives. In contrast to most classical textbooks, Ganesan and Tobiska’s work is a serious attempt to combine these perspectives in a single volume. In principle, this is a very interesting and promising approach that can provide very useful insight. However, the actual conversion of the idea into the book has been done in a way that only partially achieves this goal.

From a typographical and linguistic point of view, the work is somewhat disappointing. At least in my copy, quite a few pages are printed in poor quality and are thus difficult to read. Moreover, there are many places where the readability would have greatly benefited if the quality of the English had been checked and improved by a copy editor.

The technical content is much better in most parts of the text. Indeed, the book does cover almost the full spectrum of questions that are related to finite-element methods. This starts with a thorough description of the mathematical theory, then goes on to applications in solid mechanics and fluid mechanics, and finally reaches the concrete implementation of the associated software. Unfortunately, the book is not as self-contained as the authors claim in the preface, nor is it self-explanatory (as stated on the publisher’s website). For example, some fundamental concepts are described in great detail, but others are not explained at all, so that the reader must find the information elsewhere; furthermore, there are quite a lot of instances where explanations and proofs of certain properties merely consist of a reference to some other book. But a reader who follows these links will be guided to answers to all questions and will then obtain a very broad and almost complete overview of the finite-element method. The only notable omission, in my opinion a very regrettable omission, is that there is almost no discussion at all regarding the problems that one encounters when trying to solve the very large linear equation systems that the finite-element method produces. This is an important part of the method, and a topic in which many researchers have made significant progress in recent decades, but Ganesan and Tobiska only provide a very brief three-page summary.

Nevertheless, a particularly valuable outcome for the readers is that they will learn how certain mathematical features of the method are to be interpreted in the language of the engineer. On the other hand, the method can also be derived on the basis of engineering principles, and the book also provides a translation of the associated ideas that an engineer would apply into mathematical language.

Wherever possible and reasonable, the authors have written their book in an easily understandable way, so it is certainly useful for beginners in the area. In view of its interdisciplinary nature, it is also of interest for people who have already looked into finite-element methods from one of the three standpoints that the book takes (mathematics, engineering, and computer science) and now want to learn about the other two perspectives.