A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions

Philippe Angot

doi:10.1016/j.crma.2010.04.022

Mathematical Problems in Mechanics/Partial Differential Equations

A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions
[Un modèle de domaine fictif pour le problème de Stokes/Brinkman avec des conditions de saut immergées]

Présenté par : Philippe G. Ciarlet

Philippe Angot ¹

¹ Université de Provence & LATP – CMI, UMR CNRS 6632, 39, rue F. Joliot Curie, 13453 Marseille cedex 13, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 697-702.

Résumé
Abstract

Nous présentons l'analyse d'une nouvelle méthode de domaine fictif pour des problèmes de Brinkman ou de Stokes/Brinkman permettant de traîter des conditions de sauts (J.E.B.C.) immergées générales. Notre modèle est basé sur des conditions de transmission algébriques combinant les sauts des vecteurs contrainte et vitesse sur l'interface Σ séparant deux sous-domaines. Elles sont bien choisies de façon à guarantir la coercivité de l'opérateur et issues de la généralisation à des problèmes elliptiques vectoriels d'un modèle établi dans le cas scalaire (Angot (2003, 2005) [2,3]). On prouve tout d'abord que le modèle proposé est globalement bien posé dans tout le domaine fictif et on en identifie certains sous-modèles. Une classe de méthodes est ensuite proposée dans la même formulation unifiée qui permet d'obtenir des conditions aux limites variées, comme par exemple une contrainte donnée de type Neumann ou Fourier ou une vitesse imposée sur la frontière immergée. En particulier, nous prouvons la consistance de la méthode E.B.C. pour une condition de traction imposée qui inclue la condition usuelle de sortie ouverte de l'écoulement.

We present and analyze a new fictitious domain model for the Brinkman or Stokes/Brinkman problems in order to handle general jump embedded boundary conditions (J.E.B.C.) on an immersed interface. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface Σ separating two subdomains: they are well chosen to get the coercivity of the operator. It is issued from a generalization to vector elliptic problems of a previous model stated for scalar problems with jump boundary conditions (Angot (2003, 2005) [2,3]). The proposed model is first proved to be well-posed in the whole fictitious domain and some sub-models are identified. A family of fictitious domain methods can be then derived within the same unified formulation which provides various interface or boundary conditions, e.g. a given stress of Neumann or Fourier type or a velocity Dirichlet condition. In particular, we prove the consistency of the given-traction E.B.C. method including the so-called do nothing outflow boundary condition.

Reçu le : 2009-10-05
Accepté le : 2010-04-15
Publié le : 2010-05-20

DOI : 10.1016/j.crma.2010.04.022

Affiliations des auteurs :

Philippe Angot ¹

¹ Université de Provence & LATP – CMI, UMR CNRS 6632, 39, rue F. Joliot Curie, 13453 Marseille cedex 13, France

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Philippe Angot. A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 697-702. doi : 10.1016/j.crma.2010.04.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.022/

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