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Une méthode de décomposition de domaine pour la résolution numérique d’une équation non-linéaire

Abstract : The subject of this thesis is to present a theoretical analysis and a numerical resolution of a type of quasi-linear elliptic and parabolic equations. These equations present an important role to model phenomena in population dynamics and chemical reactions. We started this thesis with the theoretical study of a quasi-linear elliptical equation for which we demonstrated the existence of a weak non-negative solution under more general hypotheses than those considered in previous works. Then we inspired a new method based on Newton’s method and the domain decomposition method without and with overlapping. Then, we recalled some theoretical aspects concerning the existence, the uniqueness and the regularity of the solution of a parabolic equation called Fujita equation. We also recalled results about the existence of the global solution and the maximum time of existence in the case of blow-up. In order to calculate a numerical approximation of the solution of this type of equation, we introduced a finite element discretization in the space variable and a Crank-Nicholson scheme for the time discretization. To solve the discrete nonlinear problem we implemented a Newton’s method coupled with a domain decomposition method. We have shown that the method is well posed. Another type of parabolic equation known as the Chipot-Weissler equation has also been treated. First, we recalled theoretical results concerning this equation. Then, based on the numerical methods studied previously, a numerical approximation of the solution of this equation was calculated. In the last section of each chapter of this thesis we presented numerical simulations illustrating the performance of the algorithms studied and its compatibility with the theory.
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  • HAL Id : tel-03128388, version 1

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Nahed Naceur. Une méthode de décomposition de domaine pour la résolution numérique d’une équation non-linéaire. Analyse numérique [math.NA]. Université de Lorraine; Université du Centre (Sousse, Tunisie), 2020. Français. ⟨NNT : 2020LORR0149⟩. ⟨tel-03128388⟩

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