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, Résumé

, Cette thèse aborde différents aspects de la résolution numérique desÉquationsdes´desÉquations aux Dérivées Partielles

, Le premier chapitre est consacréconsacréà l'´ etude de la méthode Mixed High-Order (MHO)

, Il s'agit d'une méthode mixte dedernì ere génération permettant d'obtenir des approximations d'ordre arbitraire sur maillages généraux. Le principal résultat obtenu est l'´ equivalence entre la méthode MHO et une méthode primale de type Hybrid High-Order (HHO)

. Dans-ledeuxì-eme-chapitre, HHOàHHOà desprobì emes issus de la mécanique des fluides. Nous considérons d'abord leprobì eme de Stokes, pour lequel nous obtenons une discrétisation d'ordre arbitraire inf-sup stable sur maillages généraux

, Ensuite, nousétudionsnousétudions l'extension auprobì eme d'Oseen, pour lequel on propose une estimation d'erreur en norme d'´ energie

. Dans-letroisì-eme-chapitre, Le schéma proposé permet de traiter des maillages généraux ainsi que de faire varier le degré polynomial d'unélémentàunélémentunélémentà l'autre. La dépendance de l'anisotropie locale du coefficient de diffusion est tracée explicitement dans l'

, La thèse se clôture par une ouverture sur la réduction deprobì emes de diffusionà diffusion`diffusionà coefficients variables. L'objectif consistè a comprendre l'impact du choix de la formulation (mixte ou primale) utilisée pour la projection sur l'espace réduit sur la qualité du modèle réduit

. Mots-clés, Méthodes Mixed High-Order, méthodes Hybrid High-Order, maillages généraux, analyse hp,probì eme d'Oseen,probì eme de Stokes,probì eme de Darcy