Analyse et approximation numérique de systèmes d'interaction fluide-structure et de modèles de champ-de-phase

Ce manuscrit est compose de deux parties. La premiere partie porte d'une part sur l’analyse numerique de methodes d’approximations de systemes d’interaction fluide-structure et d'autre part, sur des questions associees de controle optimal. Des resultats de convergence de schemas numeriques et d’analyse d’erreurs sont obtenus pour une structure rigide immergee dans un fluide visqueux incompressible. Les schemas numeriques etudies sont bases sur des methodes d’approximation de type Lagrange-Galerkin avec des fonctions caracteristiques specifiques pour gerer les structures solides. Des simulations numeriques sont realisees pour des structures rigides et deformables, avec en particulier une application a la nage de poissons consideree comme un phenomene d’auto-propulsion. Les questions de controle optimal sont relatives a un probleme de controle en temps minimal pour la nage a bas nombre de Reynolds. La deuxieme partie du manuscrit concerne l’analyse et l’analyse numerique de modeles de champ-de-phase pour des problemes de transitions de phase (solidification et separation de phases). Des resultats d'existence et d’unicite de solutions pour des modeles de solidification isotherme et de separation de phases ont ete obtenus. Des methodes d’approximation par Elements Finis sont etudiees avec des estimations d’erreurs a priori et a posteriori.

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