This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. 
Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. 
This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.

Eric Chung is a Professor in the Department of Mathematics and an Outstanding Fellow of the Faculty of Science at the Chinese University of Hong Kong. His research focuses on numerical discretizations of partial differential equations and the development of computational multiscale methods for challenging applications.
Yalchin Efendiev is a Professor in the Department of Mathematics at the Texas A&M University.

Thomas Y. Hou is the Charles Lee Powell Professor of Applied and Computational Mathematics at the California Institute of Technology. His research focuses on multiscale analysis and computation, fluid interface problems, and singularity formation of 3D Euler and Navier-Stokes equations.

 

 

Introduction.- Homogenization and Numerical Homogenization of Linear Equations.- Local Model Reduction: Introduction to Multiscale Finite Element Methods.- Generalized Multiscale Finite Element Methods: Main Concepts and Overview.- Adaptive Strategies.- Selected Global Formulations for GMsFEM and Energy Stable Oversampling.- GMsFEM Using Sparsity in the Snapshot Spaces.- Space-time GMsFEM.- Constraint Energy Minimizing Concepts.- Non-local Multicontinua Upscaling.- Space-time GMsFEM.- Multiscale Methods for Perforated Domains.- Multiscale Stabilization.- GMsFEM for Selected Applications.- Homogenization and Numerical Homogenization of Nonlinear Equations.- GMsFEM for Nonlinear Problems.- Nonlinear Non-local Multicontinua Upscaling.- Global-local Multiscale Model Reduction Using GMsFEM.- Multiscale Methods in Temporal Splitting. Efficient Implicit-explicit Methods for Multiscale Problems.- References.- Index.