Multiscale finite element methods : theory and applications / Yalchin Efendiev, Thomas Y. Hou

The book under review surveys the concepts and advances in multiscale finite element methods. It is intended for a broad audience, including engineers, applied scientists and those who are interested in multiscale simulations. The author combines a practical introduction, analysis of multiscale finite element methods and numerical results. The book is self-contained, starting from the basic concepts and proceeding to the latest developments in the field.
The book addresses mathematical and numerical issues in multiscale finite element methods and connects them to real-world applications. The first chapter gives a general introduction to multiscale methods and provides a key to the organization of the book and its scope. In the second chapter an overview of multiscale finite element methods is given. Extensions to nonlinear problems are discussed in the third chapter. The fourth chapter of the book is devoted to multiscale methods that use limited global information. The fifth chapter focuses on applications of the methods presented. To make the presentation accessible to a broader audience, the analyses of the methods are given in the last chapter.
Each chapter of the book starts with a simple introduction and a description of the methods proposed, as well as motivating examples. Numerical examples demonstrating the significance of the methods proposed are presented in each chapter. (MathScinet)