On a vorticity-based formulation for reaction-diffusion-Brinkman systems

Verónica Anaya; Mostafa Bendahmane; David Mora; Ricardo Ruiz Baier; Verónica Anaya; Mostafa Bendahmane; David Mora; Ricardo Ruiz Baier

doi:10.3934/nhm.2018004

Networks and Heterogeneous Media

2018, Volume 13, Issue 1: 69-94. doi: 10.3934/nhm.2018004

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On a vorticity-based formulation for reaction-diffusion-Brinkman systems

1.
GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Concepción, Chile
2.
Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33076 Bordeaux Cedex, France
3.
Centro de Investigación en Ingeniería Matemática (CI²MA), Universidad de Concepción, Concepción, Chile
4.
Mathematical Institute, University of Oxford, A. Wiles Building, Woodstock Road, Oxford OX2 6GG, UK

Received: 01 May 2017 Revised: 01 November 2017
Primary: 65M60, 35K57; Secondary: 35Q35, 76S05

We are interested in modelling the interaction of bacteria and certain nutrient concentration within a porous medium admitting viscous flow. The governing equations in primal-mixed form consist of an advection-reaction-diffusion system representing the bacteria-chemical mass exchange, coupled to the Brinkman problem written in terms of fluid vorticity, velocity and pressure, and describing the flow patterns driven by an external source depending on the local distribution of the chemical species. A priori stability bounds are derived for the uncoupled problems, and the solvability of the full system is analysed using a fixed-point approach. We introduce a primal-mixed finite element method to numerically solve the model equations, employing a primal scheme with piecewise linear approximation of the reaction-diffusion unknowns, while the discrete flow problem uses a mixed approach based on Raviart-Thomas elements for velocity, Nédélec elements for vorticity, and piecewise constant pressure approximations. In particular, this choice produces exactly divergence-free velocity approximations. We establish existence of discrete solutions and show their convergence to the weak solution of the continuous coupled problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.
- Reaction-diffusion,
- Brinkman flows,
- vorticity formulation,
- mixed finite elements,
- chemical reactions
Citation: Verónica Anaya, Mostafa Bendahmane, David Mora, Ricardo Ruiz Baier. 2018: On a vorticity-based formulation for reaction-diffusion-Brinkman systems, Networks and Heterogeneous Media, 13(1): 69-94. doi: 10.3934/nhm.2018004

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Abstract

We are interested in modelling the interaction of bacteria and certain nutrient concentration within a porous medium admitting viscous flow. The governing equations in primal-mixed form consist of an advection-reaction-diffusion system representing the bacteria-chemical mass exchange, coupled to the Brinkman problem written in terms of fluid vorticity, velocity and pressure, and describing the flow patterns driven by an external source depending on the local distribution of the chemical species. A priori stability bounds are derived for the uncoupled problems, and the solvability of the full system is analysed using a fixed-point approach. We introduce a primal-mixed finite element method to numerically solve the model equations, employing a primal scheme with piecewise linear approximation of the reaction-diffusion unknowns, while the discrete flow problem uses a mixed approach based on Raviart-Thomas elements for velocity, Nédélec elements for vorticity, and piecewise constant pressure approximations. In particular, this choice produces exactly divergence-free velocity approximations. We establish existence of discrete solutions and show their convergence to the weak solution of the continuous coupled problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.

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