Non-Fickian convection–diffusion models in porous media

• Published in: Numerische Mathematik 2018-04, Vol.138 (4), p.869-904
• Main Authors: Barbeiro, Sílvia, Bardeji, Somayeh Gh, Ferreira, José A., Pinto, Luís
• Format: Article
• Language: English
• Subjects: Approximation; Differential equations; Finite element method; Fluid pressure; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Porous materials; Porous media; Simulation; Theoretical; Transport processes
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-017-0922-6

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.