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Étude théorique et numérique des équations non-linéaires de Sobolev

Abstract : The purpose of this work is the mathematical study and the numerical analysis of the nonlinear Sobolev problem. A first chapter is devoted to the a priori analysis for the Sobolev problem, where we use an explicit semidiscretization in time. A priori error estimates were obtained ensuring that the used numerical schemes converge when the time step discretization and the spatial step discretization tend to zero. In a second chapter, we are interested in the singularly perturbed Sobolev problem. For the stability of numerical schemes, we used in this part implicit semidiscretizations in time (the Euler method and the Crank-Nicolson method). Our estimates of Chapters 1 and 2 are confirmed in the third chapter by some numerical experiments. In the last chapter, we consider a Sobolev equation and we derive a posteriori error estimates for the discretization of this equation by a conforming finite element method in space and an implicit Euler scheme in time. The upper bound is global in space and time and allows effective control of the global error. At the end of the chapter, we propose an adaptive algorithm which ensures the control of the total error with respect to a user-defined relative precision by refining the meshes adaptively, equilibrating the time and space contributions of the error. We also present numerical experiments.
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Fatiha Bekkouche. Étude théorique et numérique des équations non-linéaires de Sobolev. Equations aux dérivées partielles [math.AP]. Université de Valenciennes et du Hainaut-Cambresis, 2018. Français. ⟨NNT : 2018VALE0018⟩. ⟨tel-01861259⟩

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