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Theses

Méthodes de Volumes Finis et Singularités

Abstract : In this work, we suggest different Finite Volume methods (namely cell-center, conforming Finite Volume-Element non-conforming Finite Volume-Element methods) for problems where singularities arise. First of all, we are interested in two dimensional corner singularities that occur for some elliptic problems (Laplace problem, Stokes and Navier-Stokes systems). In fact, when we consider an elliptic problem on a non-convex domain of R², singularities can corrupt initial order of convergence of numerical methods (like Finite Difference, Finite Element or Finite Volume methods). So we show for the different methods studied therein how a scattered mesh refinement can restore optimal order of convergence. Then we deal with boundary layers that come about two dimensional singularly perturbed problems. Solution of such problems have a strong scattered gradient that cannot capture Finite Volume methods on standard grids. So we study, for a model reaction-diffusion problem, the convergence of Cell-Center, conforming Finite Volume-Element and non-conforming Finite Volume-Element methods on anisotropic meshes. Finally, we talk about singularities that occur in three dimensional elliptic problems. For the Laplace model problem, we describe these singularities. Then, we illustrate by further numerical tests how, for Cell-Center, conforming Finite Volume-Element and non-conforming Finite Volume-Element methods, a suitable refined meshes can bring about better order of convergence than uniform ones.
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https://tel.archives-ouvertes.fr/tel-00010772
Contributor : Karim Djadel <>
Submitted on : Wednesday, October 26, 2005 - 4:32:03 PM
Last modification on : Friday, November 13, 2020 - 8:44:03 AM
Long-term archiving on: : Friday, September 14, 2012 - 3:15:19 PM

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  • HAL Id : tel-00010772, version 1

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Karim Djadel. Méthodes de Volumes Finis et Singularités. Mathématiques [math]. Université de Valenciennes et du Hainaut-Cambresis, 2005. Français. ⟨tel-00010772⟩

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