Over the last decade the authors can look back to an intensive research on the approximation of eigenvalue problems especially for the transmission problems and the Schrödinger equation, frequently in connection with adaptive finite elements and integral based eigensolvers. In the list of 257 references there are pointed about twenty to each of the authors. These areas of own interest form the content of the second part of the book (comprising three chapters with 89 pages), while the first part (comprising five chapters with 184 pages) is concerned with classical fundamental topics like Laplace’s, the biharmonic and Maxwell’s eigenvalue problem which serve also for illustrating the basic theory. A ninth chapter (11 pages) is devoted to standard methods for solving matrix eigenvalue problems. The tenth and last chapter (39 pages) is entitled “Integral based eigensolvers” treating the Sukurai-Segiura method, Polozzi’s method and the recursive integral method.
The aim of the book is not only to provide the way how to apply finite elements for solving the various problems in a self-contained up-to-date treatment but also the convergence analysis including the related error estimates. As a major tool for the convergence analysis serves the Babuška-Osborn functional analytic theory which forms part of the first chapter, including proofs. The setting in this theory is the behaviour of the spectrum of compact operators under small perturbations in the operator norm. Of course, due to the complexity in many proofs, not all proofs can be completely given in the text and adequate references are provided instead. The list of references contains the names of all major players in the theory of eigenvalue approximation with, surprisingly, the exception of Philip Anselone although the notion of collective compactness is contained in the index.
A valuable feature of the book are the extensive numerical examples at the end of each of the chapters wich is concerned with the exposition of numerical methods. The authors also claim that the book can be used as a graduate textbook and make a proposal how a one-semester course can be arranged. Altogether, an experienced reader will have no problem to follow the presentation of the material and the course of the proofs. In some instances some care is in order, e.g. around the introduction of the various function spaces, where sometimes a deviation from the standard definitions is used which, as a consequence, makes it difficult to refer to properties of these spaces as stated in standard monographies.