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

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

doi:10.3934/nhm.2018004

Article Contents

2018, Volume 13, Issue 1: 69-94. Doi: 10.3934/nhm.2018004 open access

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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

* Corresponding author
* Corresponding author

Received: April 30, 2017

Revised: October 31, 2017

Published: March 2018

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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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Mathematics Subject Classification: Primary: 65M60, 35K57; Secondary: 35Q35, 76S05.

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Figure 1. Example 1. Convergence tests for the spatial (left) and temporal (right) discretisation via mixed $\mathbb{P}_1 \times \mathbb{P}_1\times\mathbb{RT}_0\times \mathbb{P}_1\times \mathbb{P}_0$ finite elements and backward Euler time stepping applied to (2.1).

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Figure 2. Example 2: snapshots at $t = 0.5$ of the bioconvection dynamics for three different regimes characterised by $\alpha = \beta = 0.1, \gamma = 41.8$ (left), $\alpha = 0.25, \beta = 2.5, \gamma = 418$ (centre), and $\alpha = \beta = 5, \gamma = 4180$ (right). Computed solutions from top to bottom: bacteria concentration, amount of oxygen, vorticity, velocity, and pressure.

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Figure 3. Example 3A: snapshots of FitzHugh-Nagumo dynamics on a porous mixture at early (left) and advanced (right) times. Computed solutions from top to bottom: membrane voltage, vorticity, and velocity.

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Figure 4. Example 3A: Number of inner Newton steps and outer Picard steps needed to reach residual convergence to a tolerance of 1e-6.

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Figure Example 3B. Approximate membrane voltage, velocity, and pressure for the FitzHugh-Nagumo dynamics on a porous mixture at early (top), moderate (middle row), and advanced (bottom panels) times.

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Figure 5. Example 4: Computed solutions (cytosolic calcium, sarcoplasmic calcium, vorticity, velocity, and pressure) for the intracellular calcium dynamics at early (left) and advanced (right) times.

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