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Schémas Volumes Finis en mécanique des fluides complexes

Abstract : This manuscript deals with the development and numerical analysis of finite volume schemes of type discrete duality (DDFV) for the discretization of the Darcy equations and the Stokes equations. A common feature of these problems, which motivates the use of DDFV schemes, is that their finite volume resolution requires to approximate all the components of the gradient of the solution. We first study the discretization of anisotropic elliptic problems with mixed Dirichlet/Fourier boundary conditions. The scheme we propose allows to build the corresponding discrete non-overlapping Schwarz algorithm associated to a decomposition of the domain, which converges to the solution of the DDFV scheme on the initial domain. Numerical experiments illustrate the theoretical results of error estimates and of the DDFV Schwarz algorithm convergence. We then propose to discretize Stokes equations with a variable viscosity. The corresponding DDFV schemes are generally illposed. To overcome this difficulty, we stabilize the mass conservation equation with different ressure terms. Secondly, we consider the case where the viscosity is discontinuous. The discontinuities must be taken into account in the scheme to overcome the consistency defect of the numerical fluxes. Then the first study of the extension of the DDFV schemes to Navier-Stokes equations is presented as well as a generalization in 3D of the results in the case of the Stokes problem with smooth variable viscosity.
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https://tel.archives-ouvertes.fr/tel-00524509
Contributor : Stella Krell <>
Submitted on : Friday, October 8, 2010 - 9:36:25 AM
Last modification on : Wednesday, October 10, 2018 - 1:26:23 AM
Long-term archiving on: : Monday, January 10, 2011 - 11:38:47 AM

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  • HAL Id : tel-00524509, version 1

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Stella Krell. Schémas Volumes Finis en mécanique des fluides complexes. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2010. Français. ⟨tel-00524509⟩

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