# 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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983. Google Scholar [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. Google Scholar [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. Google Scholar [13] A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, vol. 24 of Lecture Notes in Physics, New Series Monographs, Springer-Verlag, Heidelberg, 1994. Google Scholar [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. Google Scholar [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. Google Scholar [16] A. C. Fowler, Convective diffusion on an enzyme reaction, SIAM J. Appl. Math., 33 (1977), 289-297. doi: 10.1137/0133018. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, 1988.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reedition in the AMS-Chelsea Series, AMS, Providence, 2001. Google Scholar [37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2006. Google Scholar [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. Google Scholar

show all references

##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983. Google Scholar [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. Google Scholar [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. Google Scholar [13] A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, vol. 24 of Lecture Notes in Physics, New Series Monographs, Springer-Verlag, Heidelberg, 1994. Google Scholar [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. Google Scholar [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. Google Scholar [16] A. C. Fowler, Convective diffusion on an enzyme reaction, SIAM J. Appl. Math., 33 (1977), 289-297. doi: 10.1137/0133018. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, 1988.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reedition in the AMS-Chelsea Series, AMS, Providence, 2001. Google Scholar [37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2006. Google Scholar [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. Google Scholar
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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