Introduction

Abstract

The aim of this monograph is to describe the main concepts and recent advances in multiscale finite element methods. This monograph is intended for the broader audience including engineers, applied scientists, and for those who are interested in multiscale simulations. The book is intended for graduate students in applied mathematics and those interested in multiscale computa- tions. It combines a practical introduction, numerical results, and analysis of multiscale finite element methods. Due to the page limitation, the material has been condensed. Each chapter of the book starts with an introduction and description of the proposed methods and motivating examples. Some new techniques are introduced using formal arguments that are justified later in the last chapter. Numerical examples demonstrating the significance of the proposed methods are presented in each chapter following the description of the methods. In the last chapter, we analyze a few representative cases with the objective of demonstrating the main error sources and the convergence of the proposed methods. A brief outline of the book is as follows. The first chapter gives a general introduction tomultiscale methods and an outline of each chapter. The second chapter discusses the main idea of the multiscale finite element method and its extensions. This chapter also gives an overview of multiscale finite element methods and other related methods. The third chapter discusses the exten- sion of multiscale finite element methods to nonlinear problems. The fourth chapter focuses on multiscale methods that use limited global information. This is motivated by porous media applications where some type of nonlocal information is needed in upscaling as well as multiscale simulations. The fifth chapter of the book is devoted to applications of these methods. Finally, in the last chapter, we present analyses of some representative multiscale methods from Chapters 2, 3, and 4.