From the preface: “The purpose of this text is to offer an overview of the most popular domain decomposition methods for partial differential equations (PDEs). The presentation is kept as much as possible at an elementary level with a special focus on the definitions of these methods both in terms of PDEs and of the sparse matrices arising from their discretizations …The book is addressed to computational scientists, mathematicians, physicists, and, in general, to people involved in numerical simulation of PDEs. It can also be used as a textbook for advanced undergraduate/first-year graduate students. …We focus here on parallel linear iterative solvers. Contrary to direct methods, the appealing feature of domain decomposition methods is that they are naturally parallel. …We introduce the reader to the main classes of domain decomposition algorithms: Schwarz, Neumann-Neumann/finite elemnet tearing and interconnecting (FETI), and optimized Schwarz. For each method we start with the continuous formulation in terms of PDEs for two subdomains. We then give the definition in terms of stiffness matrices and their implementation in a free finite element package in the many-subdomain case.”
The book is an important complement to the monograph by A. Toselli and O. Widlund [Domain decomposition methods – algorithms and theory. Berlin: Springer (2005; Zbl 1069.65138)]. In line with keeping at the elementary level, the convergence analysis is left in terms of matrices. For more advanced analysis, including functional-analytic reasons for the asymptotic behavior of the methods for large problems, the reader needs to refer to literature such as the mentioned monograph by Widlund and Tosselli. One important feature of the book is the unified treatment of optimized methods. Starting from decomposition on 2 half-planes, optimal methods are developed for several problems the including biharmonic equation.
Overall, the book is accessible to students with limited theoretical background and along with the software, it can provide hands-on experience with parallel numerical solution of PDEs.
There are, however, some small omissions. The correct name of the method called balancing Neumann-Neumann here is balancing domain decomposition according to the original publication, and the method is commonly known as BDD. The most advanced and widely used versions of BDD and FETI, the BDDC and FETI-DP methods, are cited but not described. Robust coarse spaces by solving generalized eigenvalue problems on interfaces between subdomains, called GenEO here and referred to [N. Spillane et al., C. R., Math., Acad. Sci. Paris 351, No. 5–6, 197–201 (2013; Zbl 1269.65037)] and [N. Spillane and D. J. Rixen, Int. J. Numer. Methods Eng. 95, No. 11, 953–990 (2013; Zbl 1352.65553)], were introduced and analyzed earlier by the reviewer and B. Sousedík [Comput. Methods Appl. Mech. Eng. 196, No. 8, 1389–1399 (2007; Zbl 1173.74435)].