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Theses

A priori and a posteriori error analysis of mixed and non-conforming finite element methods

Abstract : In this thesis we present a priori and a posteriori error analysis of mixed and nonconforming fnite element methods. In particular, we consider the Darcy equations with variable permeability and the convection-diffusion-reaction equation in the convection-dominated regime. We discretize the Darcy equations by the mixed nonconforming Petrov-Galerkin method known as the finite volume box scheme. Residual and hierarchical techniques lead to reliable and efficient a posteriori error estimators which are applicable no matter the variation of the permeability. Finally, we present numerical results which conform the theory and we use the error indicators obtained to generate adaptive meshes. We discretize the convection-diffusion-reaction equations by nonconforming finite elements. We investigate two different methods of stabilization: the subgrid method, leading to a finite volume box scheme and the face penalty method. We prove that the derived schemes have the same convergence properties as conforming finite element approximations. Owing to the residual a posteriori error estimation techniques we obtain reliable and efficient a posteriori error estimators. Some of these estimators are robust in the sense defined by Verfürth, where the estimates are optimal if the local Peclet number is sufficiently small. Finally, to numerically illustrate the theory we consider test cases with an interior layer and we use the error indicators to generate adaptive meshes.
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Submitted on : Friday, June 3, 2005 - 8:00:00 AM
Last modification on : Friday, June 3, 2005 - 8:00:00 AM
Long-term archiving on: : Thursday, September 30, 2010 - 6:41:55 PM

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Linda El Alaoui Lakhnati. A priori and a posteriori error analysis of mixed and non-conforming finite element methods. Mathematics [math]. Ecole des Ponts ParisTech, 2005. English. ⟨pastel-00001267⟩

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