Mixed Finite Element Approximation of Reaction Front Propagation in Porous Media

Karam Allali, Siham Binna


Mixed nite element approximation of reaction front propagation model in porous media is presented. The model consists of system of reaction-diffusion equations coupled with the equations of motion under the Darcy law. The existence of solution for the semi-discrete problem is established. The stability of the fully-discrete problem is
analyzed. Optimal error estimates are proved for both semi-discrete and fully-discrete approximate schemes.

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BEAR, Jacob et VERRUIJT, Arnold. Modeling groundwater flow and pollution. Springer Science & Business Media, 1987.

WELGE, Henry J., et al. A simplified method for computing oil recovery by gas or water drive. Journal of Petroleum Technology, 1952, vol. 4, no 04, p. 91-98.

BAUMGARTNER, A. et MUTHUKUMAR, M. A trapped polymer chain in random porous media. The Journal of chemical physics, 1987, vol. 87, no 5, p. 3082-3088.

ALLALI, Karam, BELHAQ, Mohamed, et EL KAROUNI, Kamal. Influence of quasi-periodic gravitational modulation on convective instability of reaction fronts in porous media. Communications in Nonlinear Science and Numerical Simulation, 2012, vol. 17, no 4, p. 1588-1596.

ALLALI, K., DUCROT, A., TAIK, A., et al. Convective instability of reaction fronts in porous media. Mathematical Modelling of Natural Phenomena, 2007, vol. 2, no 02, p. 20-39.

AGOUZAL, A. et ALLALI, K. Numerical analysis of reaction front propagation model under Boussinesq approximation. Mathematical methods in the applied sciences, 2003, vol. 26, no 18, p. 1529-1572.

NIELD, Donald A. et BEJAN, Adrian. Convection in porous media. Springer Science & Business Media, 2006.

POLISEVSKI, Dan. The evolution Darcy-Boussinesq system (a weak maximum principle and the uniqueness). Commentationes Mathematicae Universitatis Carolinae, 1985, vol. 26, no 1, p. 181-183.

RIONERO, Salvatore. Global non-linear stability in double diffusive convection via hidden symmetries. International Journal of Non-Linear Mechanics, 2012, vol. 47, no 1, p. 61-66.

MENZINGER, M. et WOLFGANG, Re. The meaning and use of the Arrhenius activation energy. Angewandte Chemie International Edition in English, 1969, vol. 8, no 6, p. 438-444.

ZEYTOUNIAN, Radyadour Kh. Joseph Boussinesq and his approximation: a contemporary view. Comptes Rendus Mecanique, 2003, vol. 331, no 8, p. 575-586.

FORTIN, Michel et BREZZI, Franco. Mixed and hybrid finite element methods. New York: Springer-Verlag,

RAVIART, Pierre-Arnaud et THOMAS, Jean-Marie. A mixed finite element method for 2-nd order elliptic

problems. In : Mathematical aspects of finite element methods. Springer Berlin Heidelberg, 1977. p. 292-315.

BERNARDI, Christine, MADAY, Yvon, et RAPETTI, Franscesca. Discretisations variationnelles de problemes

aux limites elliptiques. Springer Science & Business Media, 2004.

CIARLET, Philippe G. The finite element method for elliptic problems. Elsevier, 1978.

MONK, Peter et MONK, Peter. Finite element methods for Maxwell’s equations. Oxford : Clarendon Press,

GRAHAM, Ivan G., SCHEICHL, Robert, et ULLMANN, Elisabeth. Mixed Finite Element analysis of

lognormal diffusion and multilevel Monte Carlo methods. arXiv preprint arXiv:1312.6047, 2013.

TEMAM, Roger. Navier-Stokes equations: theory and numerical analysis. American Mathematical Soc., 2001.

GIRAULT, Vivette et RAVIART, P.-A. Finite element approximation of the Navier-Stokes equations. Lecture

Notes in Mathematics, Berlin Springer Verlag, 1979, vol. 749.

HOLTE, JOHN M. Discrete Gronwall lemma and applications. In : MAA-NCS Meeting at the University of North Dakota. 2009.


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