Written by a very distinguished mathematician and leader in modern applied mathematics, this book is the first to provide a unified treatment of the subject, covering, in a systematic way, the general principles of multiscale models, algorithms and analysis.
People in science and engineering get familiar with multiscale modeling naturally via the multigrid algorithm for elliptic partial differential equations. However, in this book, the author presents an unusually extensive view that brings about everything from moving contact line problem to the sheet-like structures that sustain very large elastic deformations,
and from fast summation methods to domain decomposition.
This book can be divided into two parts: background materials and general topics. The background materials include an introduction to the fundamental physical models, ranging from continuum mechanics to quantum mechanics, and basic analytical techniques for multiscale problems, such as averaging methods, homogenization methods, renormalization group methods and matched asymptotics as well as classical numerical techniques that use multiscale ideas. The general topics cover examples of multi-physics models, where these are analytical models that use multi-physics coupling explicitly; numerical methods for capturing the macroscopic behavior of complex systems with the help of microscopic models, in cases when empirical macroscopic models are inadequate; numerical methods for coupling macroscopic and microscopic models locally in order to better resolve localized singularities, defects or other events. It is important to realize that multiscale modeling is not just about developing algorithms, it is also about developing better physical models.
In this book, the hierarchy of multi-physics modeling is proposed in the introduction and multiscale algorithms are discussed for elliptic equations with multiscale coefficients, problems that have multiple time scales and rare events.
This book is well-organized and combined the physical models and mathematical analysis behind the multiscale approach to understanding the essence of physical phenomena. Other books on the subject seem to concentrate on some aspect. For instance, Multiscale Finite Element Methods: Theory and Applications (Springer, 2009), by Yalchin Efendiev and Thomas Hou, emphasizes on computational methods.
This book reflects its author's very outstanding ability in mathematical analysis, physics, and scientific computing. It is the only work I have known that covers all those topics so strictly and equally in rigorous mathematical analysis, fundamental physical models and efficient numerical algorithms. This excellent book with very high quality is found in such Cambridge University Press series as the Cambridge Texts in Applied Mathematics and the Cambridge Monographs on Applied and Computational Mathematics. I think this book is uniquely suitable for those who want to study the highest achievements in this field.