Foreword
Preliminary Remarks
Contents
1 In Dimension Zero''
1.1 A Simplification to Better Understand
1.2 The Periodic Setting
1.3 Energy of Infinite Systems of Particles
1.3.1 Choice of a Model for an Infinite Periodic System
1.3.2 Introduction to (Local) Defects
1.3.3 Toward More General Perturbations
1.4 Periodicity with Defects
1.4.1 Compactly Supported Perturbations
1.4.2 Perturbation in Lp
1.4.3 An Example of Non-local Defects
1.4.4 Formalizing the Link with Systems of Particles
1.4.5 A General Deterministic Framework
1.5 The Quasi- and Almost Periodic Settings
1.5.1 The Quasi-Periodic Setting
1.5.2 The Almost Periodic Setting
1.6 The Random Setting
1.6.1 Basic Elements for the Random Setting
1.6.2 The Notion of Stationarity
1.6.3 The Periodic, Quasi-Periodic and Almost-Periodic Settings as Particular Cases of the Continuous Random Setting
1.6.4 Properties of Stationary Functions
2 Homogenization in Dimension 1
2.1 Our First One-Dimensional Cases
2.1.1 Solution and Limit of the Elliptic Equation
2.1.2 What About Numerics?
2.1.3 The Periodic Case
2.2 The Quality of Approximation: The Corrector
2.3 One-Dimensional Defects
2.4 The 1D Random Case
2.5 SomeBad'' Cases
2.5.1 The Homogenized Equation May Take a Different Form
2.5.1.1 A Simple Example
2.5.1.2 Related Examples
2.5.1.3 An Unstable Phenomenon
2.5.2 The Homogenized Equation May Not Exist, and/or be of a Different Nature
2.5.3 A Small Defect in a Specific Nonlinear Equation
3 Dimension โฅ2: The Simple'' Cases: Abstract or Periodic Settings
3.1 The Abstract Setting and Its Proof
3.1.1 An Abstract Result
3.1.2 Proof of the Abstract Result Using the Compactness Method
3.2 Interlude: When Geometry Comes into Play
3.2.1 A Laminated Material
3.2.2 Checkerboard Materials
3.2.2.1 The Periodic Checkerboard
3.2.2.2 The Random Checkerboard
3.3 Correction in the General Setting
3.3.1 Formal Intuition of the Corrector: Two-Scale Expansion
3.3.2 The Correction Theorem
3.4 Some Possible Proofs in an Explicit Case: The Periodic Setting
3.4.1 Proof in the (Very) Regular Case
3.4.2 Identification of the Homogenized Limit and Convergence via the Div-Curl Lemma
3.4.3 Convergence and Rate of Convergence
3.4.3.1 Some Preliminary Computations
3.4.3.2 A Series of Progressively Less Formal Arguments
3.4.3.3 Interior H1 Convergence
3.4.3.4 Convergence in H1 up to the Boundary
3.4.3.5 Convergence in W1,q (and OtherGradient Norms'')
3.4.4 Alternative Methods
4 Dimension โฅ2: Some Explicit Cases Beyond the Periodic Setting
4.1 Localized Defects
4.1.1 The Case of a Defect in L2(Rd)
4.1.1.1 Existence (and Uniqueness) of the Corrector
4.1.1.2 Unchanged Homogenized Coefficient
4.1.1.3 Using the Corrector
4.1.1.4 What if We Use the Periodic Corrector?
4.1.2 Case of a Defect in Lq(Rd), q=2
4.1.2.1 Genealogy of a Result
4.1.2.2 Proof of Proposition 4.1
4.1.3 A Proof of a Different Nature
4.2 Other Explicit Deterministic Cases
4.2.1 Non-local Defects
4.2.2 Quasi-Periodicity and Almost-Periodicity
4.2.2.1 Quasi-Periodicity
4.2.2.2 Almost Periodicity
4.2.3 Back to Algebras of Functions Applied to Homogenization
4.3 The Stochastic Setting
4.3.1 Existence of a Corrector
4.3.2 Back to the Particular Settings
4.3.3 Convergence to the Homogenized Problem
5 Numerical Approaches
5.1 The Classical Approach put in Action
5.1.1 Solution Procedure for an Elliptic Boundary-Value Problem Posed on a Bounded Domain
5.1.2 Application to the Homogenized Problem
5.1.3 Application to the Computation of the Periodic Corrector Function wp,per, and next of a*
5.1.4 Application to the Nonperiodic Setting
5.1.4.1 A Localized, L2, Defect Embedded in an Otherwise Periodic Structure
5.1.4.2 The Quasiperiodic Setting
5.1.5 The Random Setting
5.1.5.1 Convergence
5.1.5.2 Statistical Error and Variance Reduction
5.2 Multiscale Computational Approaches
5.2.1 One-Dimensional MsFEM
5.2.1.1 Construction of the Multiscale Approximation
5.2.1.2 Numerical Analysis
5.2.2 The MsFEM Approach in Dimension d=2
5.2.2.1 Description of the Approach
5.2.2.2 Numerical Analysis of MsFEM-lin
5.2.3 A Brief Description of HMM
5.2.4 A Variant Slightly Different in Nature
5.2.5 Some General Comments on Multiscale Approaches
5.3 Weakly Stochastic Problems
5.3.1 Randomly Deformed Periodic Structures
5.3.1.1 Mathematical Setting
5.3.1.2 Small Deformation Theory
5.3.2 Randomly Perturbed Periodic Structures
5.3.2.1 Formal Description of the Approach
5.3.2.2 Proof of the Expansion at First Order
6 Beyond the Diffusion Equation and Miscellaneous Topics
6.1 The Advection-Diffusion Equation
6.1.1 The One-Dimensional Setting
6.1.2 The General Periodic Setting
6.1.3 Beyond the Periodic Setting
6.2 Energetic Interpretation and Related Techniques
6.2.1 Rewording of the Periodic Setting in Terms of Energy Functionals
6.2.2 -Convergence Theory
6.3 Stochastic Interpretation of Deterministic Homogenization Theory
6.3.1 Deterministic Parabolic Homogenization Theory
6.3.2 A Short Reminder of the Links Between PDEs and SDEs
6.3.3 Deterministic Parabolic Homogenization Using Stochastic Processes
6.4 To Infinity... and Beyond
6.4.1 The Fully Nonlinear Setting
6.4.2 In Lieu of Conclusion
A Some Basic Elements of PDE Analysis
A.1 A Few Results in Functional Analysis
A.2 Elliptic Partial Differential Equations: Existence and Uniqueness of Solutions
A.3 Maximum Principle
A.4 Harnack Inequality
A.5 Elliptic Regularity
References
Index