<?xml version="1.0" ?>
<tei>
<teiHeader>
<fileDesc xml:id="_FrClassTSNM"/>
</teiHeader>
<text xml:lang="en">
<front> Manuscript submitted to <lb/>Website: http://AIMsciences.org <lb/>AIMS' Journals <lb/>Volume X, Number 0X, XX 200X <lb/> pp. X–XX <lb/> AN ATTEMPT AT CLASSIFYING HOMOGENIZATION-BASED NUMERICAL <lb/>MATHODS <lb/> Emmanuel Frénod <lb/> Universit´é de Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France <lb/>AND <lb/>Projet INRIA Calvi, Université de Strasbourg, IRMA, <lb/>7 rue René Descartes, F-67084 Strasbourg Cedex, France <lb/> (Communicated by the associate editor name) <lb/> Abstract. In this note, a classification of Homogenization-Based Numerical Methods and (in <lb/>particular) of Numerical Methods that are based on the Two-Scale Convergence is done. In this <lb/>classification stand: Direct Homogenization-Based Numerical Methods; H-Measure-Based Nu-<lb/>merical Methods; Two-Scale Numerical Methods and TSAPS: Two-Scale Asymptotic Preserving <lb/>Schemes. <lb/> </front>
<body> 1. Introduction. A Homogenization-Based Numerical Method is a numerical method that incor-<lb/>porates in its conception concepts coming from Homogenization Theory. Doing this gives to the <lb/>built method the capability to tackle efficiently heterogeneities or oscillations. This approach can be <lb/>applied to problems occurring in a heterogeneous medium, that have oscillating boundary conditions <lb/>or that are constrained to oscillate by an external action (for instance a magnetic field on a charged <lb/>particle cloud). <lb/>This topic is currently active. The goal of this special issue is to emphasis recent advances in this <lb/>topic in a wide variety of application fields. <lb/>This introductory paper introduces a classification of Homogenization-Based Numerical Methods, <lb/>in which stand: Direct Homogenization-Based Numerical Methods; H-Measure-Based Numerical <lb/>Methods; Two-Scale Numerical Methods and TSAPS: Two-Scale Asymptotic Preserving Schemes. <lb/>2. Direct Homogenization-Based Numerical Methods. The context of Direct Homogeniza-<lb/>tion-Based Numerical Methods is depicted in the next diagram: <lb/> u ε solution to <lb/> O ε u ε = 0 <lb/> ε → 0 <lb/> / / <lb/> u solution to <lb/> O u = 0 <lb/> u∆z solution to <lb/> O ∆z u ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> (2.1) <lb/> </body>
<front>2000 Mathematics Subject Classification. Primary: 65L99, 65M99, 65N99. <lb/> Key words and phrases. Homogenization-Based Numerical Mathods; Homogenization; Asymptotic Analysis; As-<lb/>ymptotic Expansion; Numerical Simulation. <lb/></front>
<page> 1 <lb/></page>
<note place="headnote">2 <lb/> EMMANUEL FR <lb/>ENOD <lb/></note>
<body>It is when we face with an operator O ε that generates in solution u ε of equation O ε u ε = 0 oscillations <lb/>or heterogeneities of characteristic size ε -which is small -and when it is known that, in some sense, <lb/>for small ε, u ε (z) is close to u(z) for which is known a well-posed problem O u = 0. <lb/>In this context, it is possible, in place of building a numerical approximation of operator O ε , to <lb/>build a numerical operator O ∆z approximating O. Then solving O ∆z u ∆z gives a solution u ∆z (z) <lb/>which is close to u and consequently to u ε (z), when ε is small. This approach permits to obtain an <lb/>approximation of u ε (z) without resolving the oscillations the model to compute it contains. <lb/>In the case when a corrector result is known, i.e, if in association with u(z), a function u 1 (z), <lb/>solution to well-posed equation O 1 u 1 = 0, is such that u ε (z) is close to u(z) + εu 1 (z) for small ε, it <lb/>is possible build two numerical operators O ∆z and O 1 <lb/> ∆z that are discretizations of O and O 1 . Using <lb/>them, we can compute approximated solutions u ∆z (z) and u 1 <lb/> ∆z (z) of u(z) and u 1 (z) and obtain a <lb/>good approximation of u ε (z) computing u ∆z (z) + εu 1 <lb/> ∆z (z). Such a method is called order-1 Direct <lb/>Homogenization-Based Numerical Methods and is illutrated by the following diagram. <lb/> u ε solution to <lb/> O ε u ε = 0 <lb/> ε → 0 <lb/> / / <lb/> u, u1 solutions to <lb/> O u = 0 <lb/> O 1 u 1 = 0 <lb/> u∆z, u∆z solution to <lb/> O ∆z u ∆z = 0 <lb/> O 1 <lb/> ∆z u 1 <lb/> ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> (2.2) <lb/>The paper by Legoll & Minvielle [10], by Laptev [9], by Bernard, Frénod & Rousseau [3], and by Xu <lb/>& Yue [13] of this special issue may enter this framework <lb/>3. H-Measure-Based Numerical Methods. The context of those kind of methods is when the <lb/>transition from a microscopic scale (of size ε) to a macroscopic one (of size 1) -with quantities of <lb/>interest at the microscopic scale that are not the same as the quantities of interest at the macroscopic <lb/>scale -needs to be described. This occurs for instance in the simulation of phenomena some parts <lb/>of which call upon quantum description or in the simulation of turbulence. This context can be <lb/>represented by the following diagram: <lb/> u ε solution to <lb/> O ε u ε = 0 <lb/> e ε = E(u ε ) <lb/> ε → 0 <lb/> / / <lb/> e solution to <lb/> M e = 0 <lb/> u ε <lb/> ∆z solution to <lb/> M ε <lb/> ∆z e ε <lb/> ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> ε → 0 <lb/> / / <lb/> u∆z solution to <lb/> M ∆z e ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> (3.1) <lb/>and explained as follows. The part in the top left of diagram (3.1) symbolizes a problem which is <lb/>set at the microscopic level. This problem writes O ε u ε = 0 and generates oscillations in its solution <lb/>
<note place="headnote"> CLASSIFYING HOMOGENIZATION-BASED NUMERICAL MATHODS <lb/></note>
<page>3 <lb/></page>
u ε . Besides, the quantity that makes sense at the macroscopic level is e ε ; it is related to u ε by a <lb/>non-linear relation e ε = E(u ε ) and it is, in some sense, close to e solution to M e = 0 (see the top <lb/>right of the diagram) which represents the model at the macroscopic level. <lb/>Then, the goal of a H-Measure-Based Numerical Method consists in building a numerical operator <lb/> M ε <lb/> ∆z , giving a numerical solution e ε <lb/> ∆z close to e ε , for any ε as soon as ∆z is small (see the bottom <lb/>left of the diagram), which behaves as a numerical approximation of M when ε is small (see the <lb/>bottom right of the diagram). <lb/>The paper by Tartar [12] of this special issue lays the foundation of the theory for those kinds of <lb/>methods. <lb/>4. Two-Scale Numerical Methods. The papers by Assyr, Bai & Vilmar [1], Back & Frénod [2], <lb/>Faye, Frénod & Seck [5], Frénod, Hirtoaga & Sonnendrücker [6], Lutz [11] and Henning & Ohlberger <lb/>[7] of this special issue are related to this framework of Two-Scale Numerical Methods. <lb/>An order-0 Two-Scale Numerical Method may be explained using the following diagram: <lb/> u ε solution to <lb/> O ε u ε = 0 <lb/> ε → 0 <lb/> / / <lb/> ε → 0 , two-scale ) ) <lb/> R <lb/> R <lb/>R <lb/>R <lb/>R <lb/>R <lb/>R <lb/>R <lb/> u solution to <lb/> O u = 0 <lb/> U solution to <lb/> O U = 0 <lb/> <lb/> Z <lb/> dζ <lb/> 6 6 <lb/> n <lb/>n <lb/>n <lb/>n <lb/>n <lb/>n <lb/> u∆z solution to <lb/> O ∆z u ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> U∆z solution to <lb/> O ∆z U ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> Num <lb/> Z <lb/> dζ <lb/> 6 6 <lb/> n <lb/>n <lb/>n <lb/>n <lb/>n <lb/>n <lb/> (4.1) <lb/>The context includes the one of Direct Homogenization-Based Numerical Methods and diagram (4.1) <lb/>has to be regarded as a prism. Its deepest layer is nothing but diagram (2.1). Yet, if more is known <lb/>about the asymptotic behavior of u ε , i.e. if it is known that u ε (z) is close to U (z, z <lb/> ε ), with U (z, ζ) <lb/> periodic in ζ, when ε is small (which can be translated as u ε (z) Two-Scale Converges to U (z, ζ)) and <lb/>if it is known a well posed problem O U = 0 for U (see the middle of the diagram), that gives the <lb/>equation for u (see the top right of the diagram) when integrated with respect to periodic variable <lb/> ζ, it is possible to build a specific numerical method. <lb/>This method consists in building a numerical approximation O ∆z of operator O. Using this operator <lb/>can give a numerical solution U ∆z (see the bottom of the diagram) and U ∆z (z, z <lb/> ε ) is an approxima-<lb/>tion of u ε (z) for small ε. To be consistent with the continuous level, a numerical integration of the <lb/> O ∆z U ∆z = 0 needs to yield a numerical approximation of the equation for u (see the bottom right <lb/>of the diagram). <lb/>When a little more is known concerning the asymptotic behavior of u ε when ε is small, i.e. if u ε is <lb/>close to U (z, z <lb/> ε ) + εU 1 (z, z <lb/> ε ) with U 1 (z, ζ) also periodic in ζ and if a well-posed problem is known for <lb/> U 1 , we can enrich diagram (4.1) and obtain the following diagram of order-1 Two-Scale Numerical <lb/>
<page> 4 <lb/></page>
<note place="headnote">EMMANUEL FR <lb/>ENOD <lb/></note>
Methods: <lb/> u ε solution to <lb/> O ε u ε = 0 <lb/> ε → 0 <lb/> / / <lb/> ε → 0 , two-scale ( ( <lb/> Q <lb/> Q <lb/>Q <lb/>Q <lb/>Q <lb/>Q <lb/>Q <lb/>Q <lb/>Q <lb/>Q <lb/> u, u1 solutions to <lb/> O u = 0 <lb/> O 1 u 1 = 0 <lb/> U , U 1 solutions to <lb/> O U = 0 <lb/> O 1 U 1 = 0 <lb/> <lb/> Z <lb/> dζ <lb/> 6 6 <lb/> n <lb/>n <lb/>n <lb/>n <lb/>n <lb/>n <lb/> u∆z, u 1 <lb/> ∆z solutions to <lb/> O ∆z u ∆z = 0 <lb/> O 1 <lb/> ∆z u 1 <lb/> ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> U∆z, U 1 <lb/> ∆z solutions to <lb/> O ∆z U ∆z = 0 <lb/> O 1 <lb/> ∆z U 1 <lb/> ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> Num <lb/> Z <lb/> dζ <lb/> 6 6 <lb/> n <lb/>n <lb/>n <lb/>n <lb/>n <lb/>n <lb/> (4.2) <lb/>5. TSAPS: Two-Scale Asymptotic Preserving Schemes. To describe Two-Scale Asymptotic <lb/>Preserving Schemes, it is first needed to describe what is an Asymptotic Preserving Scheme (or <lb/>AP-Scheme in short). <lb/> u ε solution to <lb/> O ε u ε = 0 <lb/> ε → 0 <lb/> / / <lb/> u solution to <lb/> O u = 0 <lb/> u ε <lb/> ∆z solution to <lb/> O ε <lb/> ∆z u ε <lb/> ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> ε → 0 <lb/> / / <lb/> u∆z solution to <lb/> O ∆z u ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> ∆z → 0 <lb/> O O <lb/> (5.1) <lb/>For this, we comment on diagram (5.1). The context is when we are face-to-face with an operator <lb/> O ε which is approached, when ε is small, by another operator O which has not the same nature as <lb/> O ε . An Asymptotic Preserving Scheme to approximate problem O ε u ε = 0 (see the top left of the <lb/>diagram) is a numerical operator O ε <lb/> ∆z that gives, when solving O ε <lb/> ∆z u ε <lb/> ∆z = 0 (see the bottom right <lb/>of the diagram) a numerical solution u ε <lb/> ∆z which is close to u, with an accuracy depending on step <lb/> ∆z and not on ε. Besides, this operator needs to mimic the behavior of an numerical approximation <lb/>(see the bottom right of the diagram) of limit problem O u = 0 (see the top right of the diagram) <lb/>as ε is small. <lb/>For an introduction to this kind of method the reader is referred to Jin [8]. <lb/>
<note place="headnote"> CLASSIFYING HOMOGENIZATION-BASED NUMERICAL MATHODS <lb/></note>
<page>5 <lb/></page>
The explanation of TSAPS, will be based on the following diagram: <lb/> u ε solution to <lb/> O ε u ε = 0 <lb/> ε → 0 <lb/> / / <lb/> ε → 0 , two-scale <lb/> + + <lb/> W <lb/> W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/>W <lb/> u, u 1 solutions to <lb/> O u = 0, O 1 u 1 = 0 <lb/> U , U 1 solutions to <lb/> O U = 0 <lb/> O 1 U 1 = 0 <lb/> <lb/> Z <lb/> dζ <lb/> 6 6 <lb/> m <lb/>m <lb/>m <lb/>m <lb/>m <lb/>m <lb/>m <lb/>m <lb/>m <lb/> U ε solution to <lb/> O ε U ε = 0 <lb/> ζ = <lb/> z <lb/> ε <lb/> ^ ^ > <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/> ε → 0 <lb/> 7 7 <lb/> o <lb/>o <lb/>o <lb/>o <lb/>o <lb/>o <lb/>o <lb/> u ε <lb/> ∆z solution to <lb/> O ε <lb/> ∆z u ε <lb/> ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> ε → 0 <lb/> / / <lb/> u ∆z , u 1 <lb/> ∆z solutions to <lb/> O ∆z u ∆z = 0, O 1 <lb/> ∆z u 1 <lb/> ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> U ∆z , U 1 <lb/> ∆z solutions to <lb/> O ∆z U ∆z = 0 <lb/> O 1 <lb/> ∆z U 1 <lb/> ∆z = 0 <lb/> ∆z → 0 <lb/> O O <lb/> Num <lb/> Z <lb/> dζ <lb/> 6 6 <lb/> m <lb/>m <lb/>m <lb/>m <lb/>m <lb/>m <lb/>m <lb/>m <lb/> U ε <lb/> ∆z solution to <lb/> O ε <lb/> ∆z U ε <lb/> ∆z = 0 <lb/> ζ = <lb/> z <lb/> ε <lb/> ^ ^ > <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/>> <lb/> ∆z → 0 <lb/> O O <lb/> ε → 0 <lb/> 7 7 <lb/> o <lb/>o <lb/>o <lb/>o <lb/>o <lb/>o <lb/> (5.2) <lb/>This diagram has to be regarded as a prism with three layers. The deepest one is the diagram of <lb/>the AP-schemes. At the top left of this layer is found the equation that generates in its solution <lb/>oscillations of size ε. At the top right stands the limit problem, as ε is small. (This limit problem <lb/>is assumed to be an order-1 problem, i.e. u ε ∼ u 0 + εu 1 for ε small and equations are known for u 0 <lb/> and u 1 .) At the bottom left stands the AP-Scheme that approximate equation O ε u ε = 0 for any ε <lb/> and that mimics an approximation of the limit problem when ε is small (see the bottom right of the <lb/>layer). <lb/>The middle layer is exactly the diagram of the order-1 Two-Scale Numerical Methods. <lb/>The top layer is the new part. at the bottom, stands the TSAPS. This is a numerical method that <lb/>gives a solution U ε <lb/> ∆z which depends on two variables z and ζ. When taken in ζ = z/ε, U ε <lb/> ∆z gives <lb/>a numerical approximation of the solution to the problem given at the top right of the diagram, <lb/>with an accuracy that only depends on the discretization step ∆z, and not on ε. Moreover, as ε is <lb/>small, the TSAPS O ε <lb/> ∆z needs to mimic the behavior of the order-1 Two-Scale Numerical Operator <lb/>(the couple (O, O 1 )). To builtd a TSAPS, a reformulation of problem O ε u ε = 0 calling upon a <lb/>Two-Scale Macro-Micro Decomposition (that reads O ε U ε = 0, see the middle of the diagram) is <lb/>used. A first step towards TSAPS is led in Crouseilles, Frénod, Hirstoaga & Mouton [4]. <lb/></body>
<listBibl> REFERENCES <lb/> [1] A. Abdulle, Y. Bai, and G. Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilin-<lb/>ear elliptic homogenization problems. Discrete and Continuous Dynamical Systems -Serie S. Special Issue on <lb/>Numerical Methods based on Homogenization and Two-Scale Convergence, In press. <lb/>[2] A. Back and E. Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. <lb/> Discrete and Continuous Dynamical Systems -Serie S. Special Issue on Numerical Methods based on Homoge-<lb/>nization and Two-Scale Convergence, In press. <lb/>
<page> 6 <lb/></page>
<note place="headnote"> EMMANUEL FR <lb/>ENOD <lb/></note>
[3] J.-P. Bernard, E. Frénod, and A. Rousseau. Paralic confinement computations in coastal environment with <lb/>interlocked areas. Discrete and Continuous Dynamical Systems -Serie S. Special Issue on Numerical Methods <lb/>based on Homogenization and Two-Scale Convergence, In press. <lb/>[4] N. Crouseilles, E. Frenod, S. Hirstoaga, and A. Mouton. Two-Scale Macro-Micro decomposition of the Vlasov <lb/>equation with a strong magnetic field. Mathematical Models and Methods in Applied Sciences, 23(08):1527–1559, <lb/>November 2012. <lb/> [5] I. Faye, E. Frénod, and D. Seck. Two-scale numerical simulation of sand transport problems. Discrete and <lb/>Continuous Dynamical Systems -Serie S. Special Issue on Numerical Methods based on Homogenization and <lb/>Two-Scale Convergence, In press. <lb/>[6] E. Frénod, S. Histoaga, and E. Sonnendrücker. An exponential integrator for a highly oscillatory Vlasov equa-<lb/>tion. Discrete and Continuous Dynamical Systems -Serie S. Special Issue on Numerical Methods based on <lb/>Homogenization and Two-Scale Convergence, In press. <lb/>[7] P. Henning and M. Ohlberger. Error control and adaptivity for heterogeneous multiscale approximations of <lb/>nonlinear monotone problems. Discrete and Continuous Dynamical Systems -Serie S. Special Issue on Numerical <lb/>Methods based on Homogenization and Two-Scale Convergence, In press. <lb/>[8] S Jin. Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations. SIAM Journal of <lb/>Scientific Computing, 21:441–454, 1999. <lb/>[9] V. Laptev. Deterministic homogenization for media with barriers. Discrete and Continuous Dynamical Systems <lb/>-Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, In press. <lb/>[10] F. Legoll and W. Minvielle. Variance reduction using antithetic variables for a nonlinear convex stochastic homog-<lb/>enization problem. Discrete and Continuous Dynamical Systems -Serie S. Special Issue on Numerical Methods <lb/>based on Homogenization and Two-Scale Convergence, In press. <lb/>[11] M. Lutz. Application of Lie transform techniques for simulation of a charged particle beam. Discrete and Contin-<lb/>uous Dynamical Systems -Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale <lb/>Convergence, In press. <lb/>[12] Tartar. Multi-scales h-measures. Discrete and Continuous Dynamical Systems -Serie S. Special Issue on Nu-<lb/>merical Methods based on Homogenization and Two-Scale Convergence, In press. <lb/>[13] X. Xu, S. Yue. Homogenization of thermal-hydro-mass transfer processes. Discrete and Continuous Dynamical <lb/>Systems -Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, <lb/> In press. <lb/></listBibl>
<note place="footnote">E-mail address: emmanuel.frenod@univ-ubs.fr</note>
</text>
</tei>
zeynalig