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Méthodes d'éléments finis pour le problème de Darcy couplé avec l'équation de la chaleur

Abstract : In this thesis, we study the heat equation coupled with Darcy's law by a nonlinear viscosity depending on the temperature in dimension d=2,3 (Hooman and Gurgenci or Rashad). We analyse this problem by setting it in an equivalent variational formulation and reducing it to an diffusion-convection equation for the temperature where the velocity depends implicitly on the temperature.Existence of a solution is derived without restriction on the data by Galerkin's method and Brouwer's Fixed Point. Global uniqueness is established when the solution is slightly smoother and the dataare suitably restricted. We also introduce an alternative equivalent variational formulation. Both variational formulations are discretized by four finite element schemes in a polygonal or polyhedral domain. We derive existence, conditional uniqueness, convergence, and optimal a priori error estimates for the solutions of the three schemes. Next, these schemes are linearized by suitable convergent successive approximation algorithms. We present some numerical experiments for a model problem that confirm the theoretical rates of convergence developed in this work. A posteriori error estimates are established with two types of errors indicators related to the linearisation and discretization. Finally, we show numerical results of validation.
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https://tel.archives-ouvertes.fr/tel-01707846
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Submitted on : Tuesday, February 13, 2018 - 10:59:05 AM
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Serena Dib. Méthodes d'éléments finis pour le problème de Darcy couplé avec l'équation de la chaleur. Physique mathématique [math-ph]. Université Pierre et Marie Curie - Paris VI; Faculté des sciences religieuses de l'Université Saint-Joseph (Beyrouth, Liban), 2017. Français. ⟨NNT : 2017PA066294⟩. ⟨tel-01707846⟩

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