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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.
List of contents
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.
Report
This is a self-contained presentation of the multiscale finite element methods. Each chapter starts with motivating examples and a description of the methods. This book provides a good starting point for those interested in multiscale finite element methods. I recommend this book to any graduate students and scholars seeking to solve multiscale problems with finite element methods. (Huadong Gao, Mathematical Reviews, January, 2025)
The book is a nice survey of multiscale model reduction and is suitable for researchers in other areas who wish to approach this domain and also for specialists in the field as a general reference. (Nicolae Cîndea, zbMATH 1543.65001, 2024)
