For many years continuum mechanics elaborated models of media and methods of finding solutions on the basis of these models. The abilities of engineers to solve the resulting boundary value problems were restricted to a comparatively small set of problems either having analytic solutions or approximate solutions within reasonable amounts of computation. Computers changed not only the viewpoint from which numerical methods could be exploited, but also the choice of mathematical models relevant to applications. The set of practical models grew, as did the set of practical problems whose solutions computer found with a good accuracy. In the pre-computer era, the main problem of continuous mechanics was to formulate boundary value problems for the corresponding systems of partial differential equations; now the emphasis has shifted to the development of finite models that accurately describe space-distributed objects while taking into account the great complexity of material properties. In this sense the finite element method (FEM) has many advantages, because it embeds and inherits all the results of the continuum theory and enhances our abilities to solve new industrial problems. However, the use of the finite element method (or better, methods, since several modifications exist) is often far from the use of a “black box” program where a user can simply push a button and get a result. For accurate solution of complex problems it is necessary to understand the background of FEM and the domain of its applicability.
The book under consideration is one of the most valuable sources of such information for practitioners. Generations of engineers and researchers learned FEM techniques from Zienkiewicz’s books [for the review of the previous editions seeZbl 0435.73072,Zbl 0974.76003,Zbl 0974.76004,Zbl 0979.74002,Zbl 0979.74003]. They are classics in the area. This one, written with R. L. Taylor, summarizes the results of renowned authors. Practical and theoretical developments in FEM imply that the volumes of this book must grow from edition to edition. To a novice the book may seem huge, but an expert will be surprised at the authors’ ability to embed so much material into such a limited amount of space. The book is actually quite brilliant: it is complete, concise, and self-contained. The explanations are clear, all the topics are well-motivated, and the book should be useful even to the non-specialist. The mathematical level of the presentation is accessible to engineers, and thus the book becomes a priceless source of tools and ideas for practitioners. The book is divided into three large volumes, which could be even larger if the authors did not exclude the computer source codes. These complete codes are freely available in Internethttp://www.bh.com/companions/fem/.
The first volume deals with the principal problems of linear continuum mechanics and presents the main ideas that underlie FEM. It discusses the mathematical formulations of the problems, needed variational results, and many details of FEM. A significant portion of the first volume is devoted to problems of linear elasticity, a field in which much of the early development of FEM occurred. But the first volume treats also a variety of linear problems of mechanics, including fluid mechanics, coupled problems of mechanics, etc. This new edition covers some important recent topics such as adaptive error control, meshless and point based methods, and some others. It is worth stressing that the mathematical prerequisites are kept to minimum, so the book should be accessible to engineers and researchers equipped with a standard knowledge of mathematics.

Reviewer: L.P.Lebedev (Rostov-na-Donu)