Parameters correction of an elliptical equation using an a posteriori estimate
Résumé
In nature several phenomena touch humanity to act and control these phenomena we try to model their evolution. To simplify the study of the equations obtained, most of the time we try to use linear forms and in this way modelling errors are made which can influence the correct analysis of these phenomena. In this work we focus on the Richards' equation, where the coefficients changes their forms from a given value hs: This value is unknown then the coefficients are approximated. We prove an a priori and an a posteriori estimates on the modelling error. This estimates allows us, using local indicators, to build an adaptive algorithm to control the modelling error and automatically determine the "best" approximation of hs. Numerical results confirm the
convergence of this procedure and the interest of this approach.
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Références
Ahusborde, E., Azeiz, M., Ben Belgacem, F., Bernardi, C., 2013, Automatic simplification of Darcy's equation with pressure depend permeability. ESAIM, M2AN. Volume 47, Number 6, pp.1797- 1820.
Bernardi, Ch., Coquel, F., Raviart, P.A., 2013, Automatic coupling and finite element discretization of Navier-Stokes and heat equation, hal-0045244.
Braack, M., Ern, A., 2003, A posteriori control of modeling errors and discretization errors, Multiscale model. Simul. Vol. 1, No. 2,221-238.
Brooks, RH., Corey, AT., 1964, Hydraulic properties of porous media, Hydrology paper No. 3, Fort Collins, Colorado State University. p 27.
Hecht, F., Le Hyaric, A., Morice, J., Ohtsuka, K., Pironneau, O.: FreeFem++ http://www.freefem.org/ ff++.
Perotto, S., 2006, Adaptive modeling for free-surface flows, ESAIM, M2AN. Vol. 40, Number 3, 469-499.
Schaap, M.G., Van Genuchten, M.TH.,2006, A modified Mualem-Van Genuchten formulation for improved description of the hydraulic conductivity near saturation, Vadoze journal, 5, 25-34.
Stein, E., Ohnimus, S., 1999, Anisotropic discretization- and model-error estimation in solid mechanics by local neumann problems, Comput. Methods Appl. Mech. Engrg. 176, 363-385.
Van Genuchten, MT., 1980, A closed-form equation predicting the hydraulic conductivity of unsaturated soils, Soil sa. soc. Am. J, 44, 892-898.
Vogel, V., Van Genuchten, M.Th., Cislerova, M., 2001, Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions, Advances in water resources.24: 133-144.
Zeidler, E., 1986, Nonlinear functional analysis and its application, T1-Fixed-point theorems. Springer-verlag.New York inc.
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