Error Estimates for Discrete-Time Approximations of Nonlinear Cross-Diffusion Systems

Murakawa, Hideki

doi:10.1137/130911019

Cited by 5 publications

(1 citation statement)

References 22 publications

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“…where the function D is the so-called Kirchhoff transform (see for e.g. [25]) D i (i) := i 0 D i (r) dr, for i = c, s. Using these relations in (4.4), and recalling that the function D is strictly increasing, we get…”

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

Anaya

Bendahmane

Mora

et al. 2018

Networks &Amp; Heterogeneous Media

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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-reactiondiffusion 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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confidence: 98%