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Numerical Analysis of a New Mixed Formulation for Eigenvalue Convection-Diffusion Problems

Pierre, Charles and Plouraboué, Franck Numerical Analysis of a New Mixed Formulation for Eigenvalue Convection-Diffusion Problems. (2009) SIAM Journal on Applied Mathematics , vol. 70 (n° 3). pp. 658-676. ISSN 0036-1399

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Official URL: http://dx.doi.org/10.1137/080736442

Abstract

A mixed formulation is proposed and analyzed mathematically for coupled convection-diffusion in heterogeneous medias. Transfer in solid parts driven by pure diffusion is coupled with convection-diffusion transfer in fluid parts. This study is carried out for translation-invariant geometries (general infinite cylinders) and unidirectional flows. This formulation brings to the fore a new convection-diffusion operator, the properties of which are mathematically studied: its symmetry is first shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards −∞ and the other towards +∞, thus resulting in a nonsectorial operator. The decomposition of the convection-diffusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonsectorial, a complete solution on the infinite cylinder, associated to a step change of the wall temperature at the origin, is exhibited with the help of the operator’s two sets of eigenvalues/eigenfunctions. On the computational point of view, a mixed variational formulation is naturally associated to the eigenvalue problem. Numerical illustrations are provided for axisymmetrical situations, the convergence of which is found to be consistent with the numerical discretization.

Item Type:Article
Additional Information:Thanks to SIAM editor. This publication is available at http://epubs.siam.org/siap/
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS
Université de Toulouse > Institut National Polytechnique de Toulouse - INPT
Université de Toulouse > Université Paul Sabatier-Toulouse III - UPS
Other partners > Université de Pau et des Pays de l'Adour - UPPA (FRANCE)
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Deposited By: Franck PLOURABOUE

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