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Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations

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 diff...

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Bibliographic Details
Main Author: Frénod, Emmanuel
Format: eBook
Language:English
Published: Cham Springer International Publishing 2017, 2017
Edition:1st ed. 2017
Series:Lecture Notes in Mathematics
Subjects:
Numerical Analysis
Online Access:
Collection: Springer eBooks 2005- - Collection details see MPG.ReNa
Table of 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

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