# Systèmes multi-échelles : Modélisation et simulation by Claude Le Bris

By Claude Le Bris

Systèmes multi-échelles est une advent à l. a. problématique des systémes multi-échelles du element de vue du mathématicien appliqué. Il se compose d'une mosaique d'exemples de problèmes issus de los angeles body au sens huge qui présentent pour leur modélisation et leur simulation cette trickyé essentielle de comporter en leur sein des échelles de temps ou d'espace très différentes.

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Extra info for Systèmes multi-échelles : Modélisation et simulation (Mathématiques et Applications)

Sample text

Cependant, un point capital demeure. 42), et ce, d’abord, parce qu’on ne dispose pas non plus d’une analyse math´ematique. Certes, de r´ecents efforts de recherche dans ce domaine visent `a combler ce manque, mais dans l’´etat actuel des connaissances, on a peu d’´el´ements d’analyse, et donc on paie un in´evitable prix dans la technique num´erique. 21), on peut donner quelques ´el´ements d’analyse pour un probl`eme purement macroscopique du type inf ϕ∈A Ω W (ϕ(x), ∇ϕ(x)) dx − Ω fϕ− g ϕ. 43). Et nous allons voir que ces difficult´es sont ´enormes.

Nous prenons ici la formule de quadrature la plus simple qui consiste a` choisir un seul noeud dans chaque triangle Tj , pr´ecis´ement son barycentre ξj , et `a lui affecter le poids ωj = |Tj |. 29) se fera par cette formule de quadrature, et tout se ram`ene alors au calcul des W (ϕ)(ξj ) quand ϕ est dans l’espace d’´el´ements finis P 1 pour pouvoir ensuite l’inclure dans une boucle de minimisation sur ϕ. 28) d’o` u W (ϕ)(ξj ) = 1 2 k V (ϕ(ξj + εk) − ϕ(ξj )). 28), une mani`ere de proc´eder est de les remplacer par des sommes f (k) o` u rc est un rayon de coupure.

Et en fait, l’occurence de tel ou tel comportement d´epend non seulement de la fonctionnelle d’´energie qu’on minimise, mais aussi pr´ecis´ementde l’ensemble sur lequel on la mini2 mise. Trivialement, e−t a deux minimiseurs sur [−1, 1] mais n’en a aucun sur IR. Ceci nous conduit `a souligner un point qui a ´et´e totalement pass´e sous silence dans les sections pr´ec´edentes. 21), mais poser rigoureusement ce probl`eme requiert de pr´eciser l’espace fonctionnel o` u varie la fonction ϕ (on parle de l’espace variationnel ), ce qui est une vraie question en soi.