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On a primal-mixed, vorticity-based formulation for reaction-diffusion-Brinkman systems

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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 consist of a 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. Moreover, we establish existence of discrete solutions and show convergence to a weak solution of the original problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.

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