Numerical analysis for a chemotaxis-Navier–Stokes system

Duarte-Rodríguez, Abelardo; Rodríguez-Bellido, María Ángeles; Rueda-Gómez, Diego A.; Villamizar-Roa, Élder J.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S417-S445

In this paper we develop a numerical scheme for approximating a d-dimensional chemotaxis-Navier–Stokes system, d =2, 3, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze optimal error estimates and convergence towards regular solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.

Classification : 35Q35, 92C17, 65M12, 65M15, 65M60
Keywords: Chemotaxis-Navier–Stokes system, finite elements, convergence rates, error estimates

[1] C. Amrouche and N. E. H. Seloula, $L^p$-theory for vector potentials and Sobolev’s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions. Math. Models Methods Appl. Sci. 23 (2013) 37–92. | MR | Zbl

[2] M. Bessemoulin-Chatard and A. Jüngel, A finite volume scheme for a Keller-Segel model with additional cross-diffusion. IMA J. Numer. Anal. 34 (2014) 96–122. | MR | Zbl

[3] G. Chamoun, M. Saad and R. Talhouk, Numerical analysis of a chemotaxis-swimming bacteria model on a general triangular mesh. Appl. Numer. Math. 127 (2018) 324–348. | MR

[4] A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach. J. Fluid Mech. 694 (2012) 155–190. | MR | Zbl

[5] M. Cui, Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem. J. Comput. Appl. Math. 214 (2008) 617–636. | MR | Zbl

[6] Y. Deleuze, C.-Y. Chiang, M. Thiriet and T. W. Sheu, Numerical study of plume patterns in a chemotaxis-diffusion-convection coupling system. Comput. Fluids 126 (2016) 58–70. | MR

[7] J. Douglas, Jr. and J. E. Roberts, Numerical methods for a model for compressible miscible displacement in porous media. Math. Comput. 41 (1983) 441–459. | MR | Zbl

[8] A. Duarte-Rodríguez, L. C. F. Ferreira and E. J. Villamizar-Roa, Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Disc. Contin. Dyn. Syst. Ser. B 24 (2019) 423–447. | MR

[9] Y. Epshteyn and A. Izmirlioglu, Fully discrete analysis of a discontinuous finite element method for the Keller-Segel chemotaxis model. J. Sci. Comput. 40 (2009) 211–256. | MR | Zbl

[10] F. Filbet, A finite volume scheme for the Patlak–Keller–Segel chemotaxis model. Numer. Math. 104 (2006) 457–488. | MR | Zbl | DOI

[11] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer-Verlag (1986). | MR | Zbl | DOI

[12] F. Guillén-González and M. V. Redondo-Neble, Spatial error estimates for a finite element viscosity-splitting scheme for the Navier-Stokes equations. Int. J. Numer. Anal. Model. 10 (2013) 826–844. | MR | Zbl

[13] F. Guillén-González, M. A. Rodríguez-Bellido and D. A. Rueda-Gómez, Unconditionally energy stable fully discrete schemes for a chemo-repulsion model. Math. Comput. 88 (2019) 2069–2099. | MR | DOI

[14] F. Guillén-González, M. A. Rodríguez-Bellido and D. A. Rueda-Gómez, Study of a chemo-repulsion model with quadratic production. Part I: analysis of the continuous problem and time-discrete numerical schemes. Comput. Math. App. 80 (2020) 692–713. | MR

[15] F. Guillén-González, M. A. Rodríguez-Bellido and D. A. Rueda-Gómez, Study of a chemo-repulsion model with quadratic production. Part II: analysis of an unconditionally energy-stable fully discrete scheme. Comput. Math. App. 80 (2020) 636–652. | MR

[16] J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second order time discretization. SIAM J. Numer. Anal. 27 (1990) 353–384. | MR | Zbl | DOI

[17] N. Hill and T. Pedley, Bioconvection. Fluid Dyn. Res. 37 (2005) 1–20. | MR | Zbl | DOI

[18] J. Jiang, H. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains. Asympt. Anal. 92 (2015) 249–258. | MR

[19] A. Karimi and M. Paul, Bioconvection in spatially extended domains. Phys. Rev. E 87 (2013) 53016. | DOI

[20] J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source. Math. Models Methods Appl. Sci. 26 (2016) 2071–2109. | MR

[21] 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. | MR

[22] A. Lorz, Coupled Keller-Segel Stokes model: global existence for small initial data and blow-up delay. Commun. Math. Sci. 10 (2012) 555–574. | MR | Zbl

[23] A. Marrocco, Numerical simulation of chemotactic bacteria aggregation via mixed finite elements. ESAIM: M2AN 37 (2003) 617–630. | MR | Zbl | Numdam

[24] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system. Ann. Inst. Henri Poincaré Non Linéaire Anal. 31 (2014) 851–875. | MR | Zbl | Numdam

[25] J. Nečas, Les Méthodes Directes en Théorie des Equations Elliptiques. Editeurs Academia, Prague (1967). | MR | Zbl

[26] N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis. IMA J. Numer. Anal. 27 (2007) 332–365. | MR | Zbl

[27] N. Saito, Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Commun. Pure Appl. Anal. 11 (2012) 339–364. | MR | Zbl

[28] R. Stenberg, A technique for analysing finite element methods for viscous incompressible flow. Int. J. Numer. Methods Fluids 11 (1990) 935–948. | MR | Zbl

[29] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system. Z. Angew. Math. Phys. 66 (2015) 2555–2573. | MR

[30] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines. Proc. Nat. Acad. Sci. USA 102 (2005) 2277–2282. | Zbl

[31] M. Winkler, Global large-data solutions in a chemotaxis-(Navier–)Stokes system modelling cellular swimming in fluid drops. Commun. Part. Differ. Equ. 37 (2012) 319–351. | MR | Zbl

[32] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Arch. Ration. Mech. Anal. 211 (2014) 455–487. | MR | Zbl

[33] M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 33 (2016) 1329–1352. | MR | Numdam | Zbl

[34] J. Yang and Y. Chen, Superconvergence of a combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem. Acta Math. Appl. Sin. Eng. Ser. 27 (2011) 481–494. | MR | Zbl

[35] Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional chemotaxis-Navier–Stokes system. Disc. Contin. Dyn. Syst. Ser. B 20 (2015) 2751–2759. | MR

[36] J. Zhang, J. Zhu and R. Zhang, Characteristic splitting mixed finite element analysis of Keller-Segel chemotaxis models. Appl. Math. Comput. 278 (2016) 33–44. | MR