Combined higher order finite volume and finite element scheme for double porosity and non-linear adsorption of transport problem in porous media

  • Authors:

  • Mohammed Shuker Mahmood;Milan Hokr;Michal Lukač

  • Affiliations:

  • Department of Applied Mathematics, Mechanical Engineering Faculty, University of ilina, Slovakia;Institute of New Technologies and Applied Informatics, Faculty of Mechatronics and Interdisciplinary Engineering Studies Technical University of Liberec, Czech Republic;Department of Machine Elements and Mechanisms Transport and Manipulation Technique, Mechanical Engineering Faculty, University of ilina, Slovakia

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2011

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Abstract

In this work, a dual porosity model of reactive solute transport in porous media is presented. This model consists of a nonlinear-degenerate advection-diffusion equation including equilibrium adsorption to the reaction combined with a first-order equation for the non-equilibrium adsorption interaction processes. The numerical scheme for solving this model involves a combined high order finite volume and finite element scheme for approximation of the advection-diffusion part and relaxation-regularized algorithm for nonlinearity-degeneracy. The combined finite volume-finite element scheme is based on a new formulation developed by Eymard et al. (2010) [10]. This formulation treats the advection and diffusion separately. The advection is approximated by a second-order local maximum principle preserving cell-vertex finite volume scheme that has been recently proposed whereas the diffusion is approximated by a finite element method. The result is a conservative, accurate and very flexible algorithm which allows the use of different mesh types such as unstructured meshes and is able to solve difficult problems. Robustness and accuracy of the method have been evaluated, particularly error analysis and the rate of convergence, by comparing the analytical and numerical solutions for first and second order upwind approaches. We also illustrate the performance of the discretization scheme through a variety of practical numerical examples. The discrete maximum principle has been proved.