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Développement et analyse de méthodes de volumes finis

Abstract : This document is a synthesis of a set of works concerning the development and the analysis of finite volume methods used for the numerical approximation of partial differential equations (PDEs) stemming from physics. In the first part, the document deals with colocalized Godunov type schemes for the Maxwell and wave equations, with a study on the loss of accuracy of this scheme at low Mach number. In the second part, discrete differential operators are built on fairly general, in particular very distorted or nonconforming, bidimensional meshes. These operators are used to approach the solutions of PDEs modelling diffusion, electro and magnetostatics and electromagnetism by the discrete duality finite volume method (DDFV) on staggered meshes. The third part presents the numerical analysis and some a priori as well as a posteriori error estimations for the discretization of the Laplace equation by the DDFV scheme. The last part is devoted to the order of convergence in the L^2 norm of the finite volume approximation of the solution of the Laplace equation in one dimension and on meshes with orthogonality properties in two dimensions. Necessary and sufficient conditions, relatively to the mesh geometry and to the regularity of the data, are provided that ensure the second-order convergence of the method.
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Habilitation à diriger des recherches
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Contributor : Pascal Omnes <>
Submitted on : Wednesday, August 3, 2011 - 4:47:27 PM
Last modification on : Tuesday, October 20, 2020 - 3:56:28 PM
Long-term archiving on: : Friday, November 4, 2011 - 2:21:07 AM


  • HAL Id : tel-00613239, version 1


Pascal Omnes. Développement et analyse de méthodes de volumes finis. Mathématiques [math]. Université Paris-Nord - Paris XIII, 2010. ⟨tel-00613239⟩



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