Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations

Emmanuel Frénod is Professor of Applied Mathematics at Université Bretagne Sud.

• 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.

“This is a good research monograph for people working on theoretical and numerical aspects of oscillatory singularly perturbed differential equations. The book is well-written with several examples from various applications. This book provides the complete picture of two-scale convergence approach for homogenization problems and the numerical approach. This monograph is excellent and well-written. This book will be very useful for mathematicians and engineers working on multiscale problems.” (Srinivasan Natesan, zbMATH 1383.65084, 2018)