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Analyse a posteriori et adaptation de maillage pour des problèmes d'écoulements souterrains et à surface libre

Abstract : The main objective of this thesis is to analyze and develop effective adaptive numerical tools for the problems of underground and free-surface flow by suggesting a new adaptation method based on a posteriori error estimators.The first part sets the mathematical framework by providing a detailed analysis of the existence, the uniqueness, and the error convergence in different norms for flow equations and transportin porous media.The second part has been dedicated to a new strategy of mesh adaptation which consists of coupling two strategies ; namely (i) the method Adapt, in which a triangle is divided into four-sub-triangles ; and (ii) the strategy Newest Vertex bisection (NVB), in which a triangle isdivided by joining the middle of the longest edge to the opposite vertex. The conformity of our method emanates from the latter.The objective of the third part is to validate our new strategy. The finite volume scheme "vertex centered" has been then used in consideration of the second-order elliptic equation with discontinuous coefficients to take into account the heterogeneities. The effectiveness of themethod is proven and confirmed by numerical tests, as well as the convergence of the exact error and the estimator ; the ratio of which, when close to 1, defines the efficiency index.In the fourth part, the adaptation strategy has been optimized by designing a multi-level strategy which led to a new numerical method denominated finite volume-semi lagrangian. Its principle is to solve problems by a finite volume phase (corrector phase) preceded by a Lagrangian phase using the characteristics method (predictor phase). The former phase uses a numerical flow at the interfaces. The true physical flow is evaluated in an approximated state at these interfaces. It is obtained during the predictor phase by the characteristics method which is also based on judiciously chosen interpolation processes.
Keywords : Vertex centered
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Tarek Ghoudi. Analyse a posteriori et adaptation de maillage pour des problèmes d'écoulements souterrains et à surface libre. Equations aux dérivées partielles [math.AP]. Université Sorbonne Paris Cité, 2018. Français. ⟨NNT : 2018USPCD059⟩. ⟨tel-02519225⟩



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