A Course in Homogenization-Based Techniques focuses on providing practical insights into select mathematical techniques for multiscale problems, with a close eye on real-world applications. Supported by examples and exercises chosen for their ability to serve as tools illustrating underlying theories — much like appetizers designed to spark further curiosity — this textbook explores the 'why' just as much as it explains the 'how' of multiscale modeling. It encourages readers to pose their own questions and guides them in approaching this widely applicable and deeply valuable field. Asymptotic analysis — in particular homogenization-based techniques — can offer powerful approximate models and concrete expressions for constitutive laws that bridge the gap between microscopic and macroscopic scales for endless real-world scenarios. Understanding how to couple these scales correctly allows one to unlock entirely new interpretations of otherwise inaccessible microscopic data by linking it to macroscopic experimental observations.
Learning to use multiscale modeling in this way opens up a world of possibilities and introduces new dimensions of research to even experienced scientists. For this reason, this textbook seeks not only to train master's and PhD students of applied mathematics in techniques that can be both relevant for their education and of practical importance to their careers, but also to engage with the broader scientific community by enhancing the general mathematical literacy of scientists in the natural sciences and engineering.
Contents:
- Introduction
- First Homogenization Attempts: Rigorous Convergence Arguments and Formal Two-Scale Asymptotics
- Homogenization of a Linear Second-Order Elliptic Equation: A Two-Scale Convergence Approach
- Handling Porous Media as Periodically Perforated Media: The Homogenization Route
- Interplaying Limits and Preservation of Cultural Heritage: Dimension Reduction, Periodic Homogenization, and Fast-Reaction Asymptotics
- Finite Element Codes in FEniCS for a Linear Elliptic Equation Posed in a Perforated Domain and for Its Homogenized Version
- Toolbox in Applied Functional Analysis
- Appendix: FreeFEM++ Code Solving an Exercise from Chapter 6
Readership: Advanced undergraduate and graduate students in applied mathematics. Also researchers and practitioners in the fields of natural sciences and engineering where multiscale aspects are relevant.