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Analyse mathématique et approximation numérique des équations de Stokes et de Navier-Stokes avec des conditions aux limites non standard

Nour El Houda Seloula 1 
1 CONCHA - Complex Flow Simulation Codes based on High-order and Adaptive methods
Inria Bordeaux - Sud-Ouest, UPPA - Université de Pau et des Pays de l'Adour, CNRS - Centre National de la Recherche Scientifique : UMR5142
Abstract : This work of thesis deals with the solving of the Stokes problem, rst with boundary conditions on the normal component of the velocity fi eld and the tangential component of the vorticity, next with boundary conditions on the pressure and the tangential component of the velocity fi eld. In each case, we give existence, uniqueness and regularity of solutions. The case of very weak solutions is also treated by using a duality argument. The functional framework that we have choosed is that of Banach spaces of type H(div) and H(rot) or their intersection based on the space Lp , with 1 < p < ∞. In particular, we suppose that Ω is multiply connected and that the boundary Γ is not connexe. We are interested in a fi rst time by some Sobolev inequality for vector fields u ∈ Lp (Ω). In a second time, we give some results concerning vector potentials with different boundary conditions. This allow to establish Helmholtz decompositions and Inf − Sup condition when the bilinear form is a rotational product. Due to these non standard boundary conditions, the pressure is decoupled from the system. It is the reason whay we are naturally reduced to solving elliptic problems which are the Stokes equations without the pressure term. For this, we use the Inf − Sup conditions, which plays a crutial role in the existence and uniqueness of solutions. We give an application to the Navier-Stokes equations where the proof of solutions is obtained by applying a fixed point theorem over the Oseen equations. Finally, two numerical methods are proposed inorder to approximate the Stokes problem. First, by means of the Nitsche method and next by means of the Discontinuous Galerkin method. Some numerical results of convergence verifying the theoretical predictions are given
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Nour El Houda Seloula. Analyse mathématique et approximation numérique des équations de Stokes et de Navier-Stokes avec des conditions aux limites non standard. Equations aux dérivées partielles [math.AP]. Université de Pau et des Pays de l'Adour, 2010. Français. ⟨tel-00687740⟩

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