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Description: |
We are interested in the two following model problems:The first equation is a diffusion equation with variable coefficients which models many problems such as the prediction of the presence of oil or gaz in porous media. u is then the pressure of the fluid and represents the permeability of the medium which eventually admits large variations and a fine structure over the domain. The numerical difficulty is then to be able to capture these fine structures with an affordable numerical scheme. The second equation is the convection-diffusion equation corresponding to a singurlarly perturbed problem when the parameter is very small. Capturing the solution inaccurately can generate spurious oscillations in the computed solution. The multi-scale finite element method has been initially proposed by I. Babuska [1] : it consists in incorporating the fine scales in the macro scale basis functions by resolving the homogeneous original differential equations on each element with adapted boundary conditions. This paper proposes a new approach for the multiscale finite element method [4, 3, 2] : it consists in capturing the so-called "fine scale" using the reduced basis method [5]. This approach allows to build new basis functions at reduced cost -- with respect to the original formulation -- associated to the multi-scale problem and in particular permits to capture or control the oscillations in the fine scales at macro-scale level. [1] I. Babuska and E. J. Osborn. Generalized finite element methods: Their performances and their relation to mixed methods. SIAM J. Numer. Anal., 20:510536, 1983. [2] F. Brezzi. Subgrid scales, augmented problems, and stabilizations. Computational Fluid and Solid Mechanics, pages 8-10, 2001. [3] T. J. R. Hughes F. Brezzi, L. P. Franca and A. Russo. b = g. Appl. Mech. Engrg. Comput. Methods, 145(3-4):329-339, 1997. [4] T. Y. Hou and X. Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134(1):169189, 1997. [5] C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A. T. Patera, and G. Turinici. Reliable real- time solution of parametrized partial differential equations: Reduced-basis output bound methods. Journal of Fluids Engineering, 124(1):70-80, March 2002.
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Date: |
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| Start Time: |
11:30 |
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Speaker: |
Christophe Prud'homme (Université de Grenoble, France)
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Institution: |
Université de Grenoble
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Place: |
Room 5.5
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| Research Groups: |
-Numerical Analysis and Optimization
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