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Non-Fickian convection–diffusion models in porous media

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Abstract

In this paper we propose a numerical scheme to approximate the solution of a non-Fickian coupled model that describes, e.g., miscible transport in porous media. The model is defined by a system of a quasilinear elliptic equation, which governs the fluid pressure, and a quasilinear integro-differential equation, which models the convection–diffusion transport process. The numerical scheme is based on a conforming piecewise linear finite element method for the discretization in space. The fully discrete approximations is obtained with an implicit–explicit method. Estimates for the continuous in time and the fully discrete methods are derived, showing that the numerical approximation for the concentrations and the pressure are second order convergent in a discrete \(L^2\)-norm and in a discrete \(H^1\)-norm, respectively.

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, EC Universidade, 3001-454, Coimbra, Portugal

    Sílvia Barbeiro, Somayeh Gh. Bardeji, José A. Ferreira & Luís Pinto

  2. Medical Imaging Research Center, Shiraz University of Medical Sciences, Shiraz, Iran

    Somayeh Gh. Bardeji

Authors
  1. Sílvia Barbeiro
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  2. Somayeh Gh. Bardeji
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  3. José A. Ferreira
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  4. Luís Pinto
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Corresponding author

Correspondence to José A. Ferreira.

Additional information

This work was partially supported by the Centre for Mathematics of the University of Coimbra—UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. L. Pinto was also supported by FCT scholarship SFRH/BPD/112687/2015. S. Barbeiro also acknowledges the support of the Fundação para a Ciência e a Tecnologia, I.P.

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Barbeiro, S., Bardeji, S.G., Ferreira, J.A. et al. Non-Fickian convection–diffusion models in porous media. Numer. Math. 138, 869–904 (2018). https://doi.org/10.1007/s00211-017-0922-6

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  • DOI: https://doi.org/10.1007/s00211-017-0922-6

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