# American Institute of Mathematical Sciences

March  2018, 13(1): 69-94. doi: 10.3934/nhm.2018004

## 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

Received  May 2017 Revised  November 2017 Published  March 2018

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.

Citation: Verónica Anaya, Mostafa Bendahmane, David Mora, Ricardo Ruiz Baier. On a vorticity-based formulation for reaction-diffusion-Brinkman systems. Networks and Heterogeneous Media, 2018, 13 (1) : 69-94. doi: 10.3934/nhm.2018004
##### References:
 [1] A. Agosti, L. Formaggia and A. Scotti, Analysis of a model for precipitation and dissolution coupled with a Darcy flux, J. Math. Anal. Appl., 431 (2015), 752-781.  doi: 10.1016/j.jmaa.2015.06.003. [2] A. Agouzal and K. Allali, Numerical analysis of reaction front propagation model under Boussinesq approximation, Math. Meth. Appl. Sci., 26 (2003), 1529-1572.  doi: 10.1002/mma.425. [3] V. Anaya, G. N. Gatica, D. Mora and R. Ruiz-Baier, An augmented velocity-vorticity-pressure formulation for the Brinkman equations, Int. J. Numer. Methods Fluids, 79 (2015), 109-137.  doi: 10.1002/fld.4041. [4] V. Anaya, D. Mora, R. Oyarzúa and R. Ruiz-Baier, A priori and a posteriori error analysis of a fully-mixed scheme for the Brinkman problem, Numer. Math., 133 (2016), 781-817.  doi: 10.1007/s00211-015-0758-x. [5] V. Anaya, D. Mora, C. Reales and R. Ruiz-Baier, Stabilized mixed approximation of axisymmetric Brinkman flows, ESAIM: Math. Model. Numer. Anal., 49 (2015), 855-874.  doi: 10.1051/m2an/2015011. [6] V. Anaya, D. Mora and R. Ruiz-Baier, Pure vorticity formulation and Galerkin discretization for the Brinkman equations, IMA J. Numer. Anal., 37 (2017), 2020-2041.  doi: 10.1093/imanum/drw056. [7] J.-L. Auriault, On the domain of validity of Brinkman's equation, Transp. Porous Med., 79 (2009), 215-223.  doi: 10.1007/s11242-008-9308-7. [8] J. W. Barret and P. Knabner, Finite element approximation of the transport of reactive solutes in porous media. Part Ⅱ: error estimates for equilibrium adsorption processes, SIAM J. Numer. Anal., 34 (1997), 455-479.  doi: 10.1137/S0036142993258191. [9] P. Biscari, S. Minisini, D. Pierotti, G. Verzini and P. Zunino, Controlled release with finite dissolution rate, SIAM J. Appl. Math., 71 (2011), 731-752.  doi: 10.1137/100790288. [10] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983. [11] G. Chamoun, M. Saad and R. Talhouk, A coupled anisotropic chemotaxis-fluid model: The case of two-sidedly degenerate diffusion, Comput. Math. Appl., 68 (2014), 1052-1070.  doi: 10.1016/j.camwa.2014.04.010. [12] C. M. Elliott and B. Stinner, A surface phase field model for two-phase biological membranes, SIAM J. Appl. Math., 70 (2010), 2904-2928.  doi: 10.1137/090779917. [13] A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, vol. 24 of Lecture Notes in Physics, New Series Monographs, Springer-Verlag, Heidelberg, 1994. [14] A. Ern and J. L, Guermond and L. Quartapelle, Vorticity-velocity formulations of the Stokes problem in 3D, Math. Methods Appl. Sci., 22 (1999), 531-546.  doi: 10.1002/(SICI)1099-1476(199904)22:6<531::AID-MMA51>3.0.CO;2-9. [15] L. Formaggia, S. Minisini and P. Zunino, Modelling polymeric controlled drug release and transport phenomena in the arterial tissue, Math. Models Methods Appl. Sci., 20 (2010), 1759-1786.  doi: 10.1142/S0218202510004787. [16] A. C. Fowler, Convective diffusion on an enzyme reaction, SIAM J. Appl. Math., 33 (1977), 289-297.  doi: 10.1137/0133018. [17] G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications, Springer Briefs in Mathematics, Springer, Cham Heidelberg New York Dordrecht London, 2014. [18] A. Goldbeter, G. Dupont and M. J. Berridge, Minimal model for signal-induced $\mathrm{Ca}^{2+}$ oscillations and for their frequency encoding through protein phosphorylation, Proc. Natl. Acad. Sci. USA, 87 (1990), 1461-1465.  doi: 10.1073/pnas.87.4.1461. [19] Q. Hong and J. Krauss, Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the Brinkman problem, SIAM J. Numer. Anal., 54 (2016), 2750-2774.  doi: 10.1137/14099810X. [20] K. Kumar, I. S. Pop and F. A. Radu, Convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media, Numer. Math., 127 (2014), 715-749.  doi: 10.1007/s00211-013-0601-1. [21] T. Kuusi, L. Monsaingeon and J. Videman, Systems of partial differential equations in porous medium, Nonl. Anal., 133 (2016), 79-101.  doi: 10.1016/j.na.2015.11.015. [22] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, 1988. [23] H. G. Lee and J. Kim, Numerical investigation of falling bacterial plumes caused by bioconvection in a three-dimensional chamber, Eur. J. Mech. B/Fluids, 52 (2015), 120-130.  doi: 10.1016/j.euromechflu.2015.03.002. [24] P. Lenarda, M. Paggi and R. Ruiz Baier, Partitioned coupling of advection-diffusion-reaction systems and Brinkman flows, J. Comput. Phys., 344 (2017), 281-302.  doi: 10.1016/j.jcp.2017.05.011. [25] H. Murakawa, Error estimates for discrete-time approximations of nonlinear cross-diffusion systems, SIAM J. Numer. Anal., 52 (2014), 955-974.  doi: 10.1137/130911019. [26] C. Nagaiah, S. Rüdiger, G. Warnecke and M. Falcke, Adaptive numerical simulation of intracellular calcium dynamics using domain decomposition methods, Appl. Numer. Math., 58 (2008), 1658-1674.  doi: 10.1016/j.apnum.2007.10.003. [27] F. A. Radu and I. S. Pop, Newton method for reactive solute transport with equilibrium sorption in porous media, J. Comput. and Appl. Math., 234 (2010), 2118-2127.  doi: 10.1016/j.cam.2009.08.070. [28] R. Ruiz-Baier, Primal-mixed formulations for reaction-diffusion systems on deforming domains, J. Comput. Phys., 299 (2015), 320-338.  doi: 10.1016/j.jcp.2015.07.018. [29] R. Ruiz-Baier, A. Gizzi, S. Rossi, C. Cherubini, A. Laadhari, S. Filippi and A. Quarteroni, Mathematical modeling of active contraction in isolated cardiomyocytes, Math. Medicine Biol., 31 (2014), 259-283.  doi: 10.1093/imammb/dqt009. [30] R. Ruiz-Baier and I. Lunati, Mixed finite element -discontinuous finite volume element discretization of a general class of multicontinuum models, J. Comput. Phys., 322 (2016), 666-688.  doi: 10.1016/j.jcp.2016.06.054. [31] B. Saad and M. Saad, A combined finite volume-nonconforming finite element scheme for compressible two phase flow in porous media, Numer. Math., 129 (2015), 691-722.  doi: 10.1007/s00211-014-0651-z. [32] J. N. Shadid, R. S. Tuminaro and H. F. Walker, An inexact Newton method for fully coupled solution of the Navier-Stokes equations with heat and mass transport, J. Comput. Phys., 137 (1997), 155-185.  doi: 10.1006/jcph.1997.5798. [33] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360. [34] M. Slodicka, A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media, SIAM J. Sci. Comput., 23 (2002), 1593-1614.  doi: 10.1137/S1064827500381860. [35] G. Tauriello and P. Koumoutsakos, Coupling remeshed particle and phase field methods for the simulation of reaction-diffusion on the surface and the interior of deforming geometries, SIAM J. Sci. Comput., 35 (2013), B1285-B1303.  doi: 10.1137/130906441. [36] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reedition in the AMS-Chelsea Series, AMS, Providence, 2001. [37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2006. [38] P. Tracqui and J. Ohayon, An integrated formulation of anisotropic force-calcium relations driving spatio-temporal contractions of cardiac myocytes, Phil. Trans. Royal Soc. London A, 367 (2009), 4887-4905.  doi: 10.1098/rsta.2009.0149.

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##### References:
 [1] A. Agosti, L. Formaggia and A. Scotti, Analysis of a model for precipitation and dissolution coupled with a Darcy flux, J. Math. Anal. Appl., 431 (2015), 752-781.  doi: 10.1016/j.jmaa.2015.06.003. [2] A. Agouzal and K. Allali, Numerical analysis of reaction front propagation model under Boussinesq approximation, Math. Meth. Appl. Sci., 26 (2003), 1529-1572.  doi: 10.1002/mma.425. [3] V. Anaya, G. N. Gatica, D. Mora and R. Ruiz-Baier, An augmented velocity-vorticity-pressure formulation for the Brinkman equations, Int. J. Numer. Methods Fluids, 79 (2015), 109-137.  doi: 10.1002/fld.4041. [4] V. Anaya, D. Mora, R. Oyarzúa and R. Ruiz-Baier, A priori and a posteriori error analysis of a fully-mixed scheme for the Brinkman problem, Numer. Math., 133 (2016), 781-817.  doi: 10.1007/s00211-015-0758-x. [5] V. Anaya, D. Mora, C. Reales and R. Ruiz-Baier, Stabilized mixed approximation of axisymmetric Brinkman flows, ESAIM: Math. Model. Numer. Anal., 49 (2015), 855-874.  doi: 10.1051/m2an/2015011. [6] V. Anaya, D. Mora and R. Ruiz-Baier, Pure vorticity formulation and Galerkin discretization for the Brinkman equations, IMA J. Numer. Anal., 37 (2017), 2020-2041.  doi: 10.1093/imanum/drw056. [7] J.-L. Auriault, On the domain of validity of Brinkman's equation, Transp. Porous Med., 79 (2009), 215-223.  doi: 10.1007/s11242-008-9308-7. [8] J. W. Barret and P. Knabner, Finite element approximation of the transport of reactive solutes in porous media. Part Ⅱ: error estimates for equilibrium adsorption processes, SIAM J. Numer. Anal., 34 (1997), 455-479.  doi: 10.1137/S0036142993258191. [9] P. Biscari, S. Minisini, D. Pierotti, G. Verzini and P. Zunino, Controlled release with finite dissolution rate, SIAM J. Appl. Math., 71 (2011), 731-752.  doi: 10.1137/100790288. [10] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983. [11] G. Chamoun, M. Saad and R. Talhouk, A coupled anisotropic chemotaxis-fluid model: The case of two-sidedly degenerate diffusion, Comput. Math. Appl., 68 (2014), 1052-1070.  doi: 10.1016/j.camwa.2014.04.010. [12] C. M. Elliott and B. Stinner, A surface phase field model for two-phase biological membranes, SIAM J. Appl. Math., 70 (2010), 2904-2928.  doi: 10.1137/090779917. [13] A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, vol. 24 of Lecture Notes in Physics, New Series Monographs, Springer-Verlag, Heidelberg, 1994. [14] A. Ern and J. L, Guermond and L. Quartapelle, Vorticity-velocity formulations of the Stokes problem in 3D, Math. Methods Appl. Sci., 22 (1999), 531-546.  doi: 10.1002/(SICI)1099-1476(199904)22:6<531::AID-MMA51>3.0.CO;2-9. [15] L. Formaggia, S. Minisini and P. Zunino, Modelling polymeric controlled drug release and transport phenomena in the arterial tissue, Math. Models Methods Appl. Sci., 20 (2010), 1759-1786.  doi: 10.1142/S0218202510004787. [16] A. C. Fowler, Convective diffusion on an enzyme reaction, SIAM J. Appl. Math., 33 (1977), 289-297.  doi: 10.1137/0133018. [17] G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications, Springer Briefs in Mathematics, Springer, Cham Heidelberg New York Dordrecht London, 2014. [18] A. Goldbeter, G. Dupont and M. J. Berridge, Minimal model for signal-induced $\mathrm{Ca}^{2+}$ oscillations and for their frequency encoding through protein phosphorylation, Proc. Natl. Acad. Sci. USA, 87 (1990), 1461-1465.  doi: 10.1073/pnas.87.4.1461. [19] Q. Hong and J. Krauss, Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the Brinkman problem, SIAM J. Numer. Anal., 54 (2016), 2750-2774.  doi: 10.1137/14099810X. [20] K. Kumar, I. S. Pop and F. A. Radu, Convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media, Numer. Math., 127 (2014), 715-749.  doi: 10.1007/s00211-013-0601-1. [21] T. Kuusi, L. Monsaingeon and J. Videman, Systems of partial differential equations in porous medium, Nonl. Anal., 133 (2016), 79-101.  doi: 10.1016/j.na.2015.11.015. [22] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, 1988. [23] H. G. Lee and J. Kim, Numerical investigation of falling bacterial plumes caused by bioconvection in a three-dimensional chamber, Eur. J. Mech. B/Fluids, 52 (2015), 120-130.  doi: 10.1016/j.euromechflu.2015.03.002. [24] P. Lenarda, M. Paggi and R. Ruiz Baier, Partitioned coupling of advection-diffusion-reaction systems and Brinkman flows, J. Comput. Phys., 344 (2017), 281-302.  doi: 10.1016/j.jcp.2017.05.011. [25] H. Murakawa, Error estimates for discrete-time approximations of nonlinear cross-diffusion systems, SIAM J. Numer. Anal., 52 (2014), 955-974.  doi: 10.1137/130911019. [26] C. Nagaiah, S. Rüdiger, G. Warnecke and M. Falcke, Adaptive numerical simulation of intracellular calcium dynamics using domain decomposition methods, Appl. Numer. Math., 58 (2008), 1658-1674.  doi: 10.1016/j.apnum.2007.10.003. [27] F. A. Radu and I. S. Pop, Newton method for reactive solute transport with equilibrium sorption in porous media, J. Comput. and Appl. Math., 234 (2010), 2118-2127.  doi: 10.1016/j.cam.2009.08.070. [28] R. Ruiz-Baier, Primal-mixed formulations for reaction-diffusion systems on deforming domains, J. Comput. Phys., 299 (2015), 320-338.  doi: 10.1016/j.jcp.2015.07.018. [29] R. Ruiz-Baier, A. Gizzi, S. Rossi, C. Cherubini, A. Laadhari, S. Filippi and A. Quarteroni, Mathematical modeling of active contraction in isolated cardiomyocytes, Math. Medicine Biol., 31 (2014), 259-283.  doi: 10.1093/imammb/dqt009. [30] R. Ruiz-Baier and I. Lunati, Mixed finite element -discontinuous finite volume element discretization of a general class of multicontinuum models, J. Comput. Phys., 322 (2016), 666-688.  doi: 10.1016/j.jcp.2016.06.054. [31] B. Saad and M. Saad, A combined finite volume-nonconforming finite element scheme for compressible two phase flow in porous media, Numer. Math., 129 (2015), 691-722.  doi: 10.1007/s00211-014-0651-z. [32] J. N. Shadid, R. S. Tuminaro and H. F. Walker, An inexact Newton method for fully coupled solution of the Navier-Stokes equations with heat and mass transport, J. Comput. Phys., 137 (1997), 155-185.  doi: 10.1006/jcph.1997.5798. [33] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360. [34] M. Slodicka, A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media, SIAM J. Sci. Comput., 23 (2002), 1593-1614.  doi: 10.1137/S1064827500381860. [35] G. Tauriello and P. Koumoutsakos, Coupling remeshed particle and phase field methods for the simulation of reaction-diffusion on the surface and the interior of deforming geometries, SIAM J. Sci. Comput., 35 (2013), B1285-B1303.  doi: 10.1137/130906441. [36] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reedition in the AMS-Chelsea Series, AMS, Providence, 2001. [37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2006. [38] P. Tracqui and J. Ohayon, An integrated formulation of anisotropic force-calcium relations driving spatio-temporal contractions of cardiac myocytes, Phil. Trans. Royal Soc. London A, 367 (2009), 4887-4905.  doi: 10.1098/rsta.2009.0149.
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).
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.
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.
Example 3A: Number of inner Newton steps and outer Picard steps needed to reach residual convergence to a tolerance of 1e-6.
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.
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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