Full Description
This book presents the classical results of the two-scale convergence theory and explains - using several figures - why it works. It then shows how to use this theory to homogenize ordinary differential equations with oscillating coefficients as well as oscillatory singularly perturbed ordinary differential equations. In addition, it explores the homogenization of hyperbolic partial differential equations with oscillating coefficients and linear oscillatory singularly perturbed hyperbolic partial differential equations. Further, it introduces readers to the two-scale numerical methods that can be built from the previous approaches to solve oscillatory singularly perturbed transport equations (ODE and hyperbolic PDE) and demonstrates how they can be used efficiently. This book appeals to master's and PhD students interested in homogenization and numerics, as well as to the Iter community.
Contents
I Two-Scale Convergence.- 1 Introduction.- 1.1 First Statements on Two-Scale Convergence.- 1.2 Two-Scale Convergence and Homogenization.- 1.2.1 How Homogenization Led to the Concept of Two-Scale Convergence.- 1.2.2 A Remark Concerning Periodicity.- 1.2.3 A Remark Concerning Weak-* Convergence.- 2 Two-Scale Convergence - Definition and Results.- 2.1 Background Material on Two-Scale Convergence.- 2.1.1 Definitions.- 2.1.2 Link with Weak Convergence.- 2.2 Two-Scale Convergence Criteria.- 2.2.1 Injection Lemma.- 2.2.2 Two-Scale Convergence Criterion.- 2.2.3 Strong Two-Scale Convergence Criterion.- 3 Applications.- 3.1 Homogenization of ODE.- 3.1.1 Textbook Case, Setting and Asymptotic Expansion.- 3.1.2 Justification of Asymptotic Expansion Using Two-Scale Convergence.- 3.2 Homogenization of Singularly-Perturbed ODE.- 3.2.1 Equation of Interest and Setting.- 3.2.2 Asymptotic Expansion Results.- 3.2.3 Asymptotic Expansion Calculations.- 3.2.4 Justification Using Two-Scale Convergence I: Results.- 3.2.5 Justification Using Two-Scale Convergence II: Proofs.- 3.3 Homogenization of Hyperbolic PDE.- 3.3.1 Textbook Case and Setting.- 3.3.2 Order-0 Homogenization.- 3.3.3 Order-1 Homogenization.- 3.4 Homogenization of Singularly-Perturbed Hyperbolic PDE.- 3.4.1 Equation of Interest and Setting.- 3.4.2 An a Priori Estimate.- 3.4.3 Weak Formulation with Oscillating Test Functions.- 3.4.4 Order-0 Homogenization - Constraint.- 3.4.5 Order-0 Homogenization - Equation for V.- 3.4.6 Order-1 Homogenization - Preparations: Equations for U and u.- 3.4.7 Order-1 Homogenization - Strong Two-Scale Convergence of u".- 3.4.8 Order-1 Homogenization - The Function W1.- 3.4.9 Order-1 Homogenization - A Priori Estimate and Convergence.- 3.4.10 Order-1 Homogenization - Constraint.- 3.4.11 Order-1 Homogenization - Equation for V1.- 3.4.12 Concerning Numerics.- II Two-Scale Numerical Methods.- 4 Introduction.- 5 Two-Scale Method for Object Drift with Tide.- 5.1 Motivation and Model.- 5.1.1 Motivation.- 5.1.2 Model of Interest.- 5.2 Two-Scale Asymptotic Expansion.- 5.2.1 Asymptotic Expansion.- 5.2.2 Discussion.- 5.3 Two-Scale Numerical Method.- 5.3.1 Construction of the Two-Scale Numerical Method.- 5.3.2 Validation of the Two-Scale Numerical Method.- 6 Two-Scale Method for Beam.- 6.1 Some Words About Beams and Model of Interest.- 6.1.1 Beams.- 6.1.2 Equations of Interest.- 6.1.3 Two-Scale Convergence.- 6.2 Two-Scale PIC Method.- 6.2.1 Formulation of the Two-Scale Numerical Method.- 6.2.2 Numerical Results.
