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2018, 17(2): 429-448. doi: 10.3934/cpaa.2018024

## Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model

 1 Univ. Lille, CNRS, UMR 8524 -Laboratoire Paul Painlevé, F-59000 Lille, France 2 Unité de recherche : Multifractals et Ondelettes, FSM, University of Monastir, 5019 Monastir, Tunisia 3 FSEGN, University of Carthage, 8000 Nabeul, Tunisia

* Corresponding author

Received  June 2016 Revised  September 2017 Published  March 2018

In this paper, we construct a fully discrete numerical scheme for approximating a two-dimensional multiphasic incompressible fluid model, also called the Kazhikhov-Smagulov model. We use a first-order time discretization and a splitting in time to allow us the construction of an hybrid scheme which combines a Finite Volume and a Finite Element method. Consequently, at each time step, one only needs to solve two decoupled problems, the first one for the density and the second one for the velocity and pressure. We will prove the stability of the scheme and the convergence towards the global in time weak solution of the model.

Citation: Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
##### References:
 [1] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and Its Applications, 22, North-Holland, Publishing Co. , Amesterdam, 1990. [2] D. Bresch, E. H. Essoufi, M. Sy, Effects of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397. [3] R. C. Cabrales, F. Guillén-González, J. V. Gutiérrez-Santacreu, Stability and convergence for a complete model of mass diffusion, Applied Numerical Mathematics, 61 (2011), 1161-1185. [4] X. Cai, L. Liao, Y. Sun, Global regularity for the initial value problem of a 2-D Kazhikhov-Smagulov type model, Nonlinear Analysis, 75 (2012), 5975-5983. [5] X. Cai, L. Liao, Y. Sun, Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model, Discrete and Continuous Dynamical Systems, Series S, 7 (2014), 917-923. [6] C. Calgaro, E. Chane-Kane, E. Creusé, T. Goudon, $L^∞$-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios, J. Comput. Physics, 229 (2010), 6027-6046. [7] C. Calgaro, E. Creusé, T. Goudon, An hybrid finite volume-finite element method for variable density incompressible flows, J. Comput. Physics, 227 (2008), 4671-4696. [8] C. Calgaro, E. Creusé, T. Goudon, Modeling and simulation of mixture flows: Application to powder-snow avalanches, Computers and Fluids, 107 (2015), 100-122. [9] C. Calgaro, M. Ezzoug, $L^∞$-stability of IMEX-BDF2 finite volume scheme for convection-diffusion equation, Finite Volumes for Complex Applications Ⅷ -Methods and Theoretical Aspects, 2 (2017), 245-253. [10] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1979. [11] J. Droniou, Finite volume schemes for diffusion equations: Introduction to and review of modern methods, Mathematical Models and Methods in Applied Sciences, 24 (2014), 1575-1619. [12] J. Étienne, P. Saramito, A priori error estimates of the Lagrange-Galerkin method for Kazhikhov-Smagulov type systems, C.R. Acad. Sci. Paris Ser. I, 341 (2005), 769-774. [13] R. Eymard, T. Gallouët, R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, vol. Ⅶ, North-Holland, Amsterdam, (2000), 713-1020. [14] M. Feistauer, J. Felcman, M. Lukáčová-Medvid'ová, On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems, Numerical Methods Partial Differential Equations, 13 (1997), 163-190. [15] M. Feistauer, J. Felcman, M. Lukáčová-Medvid'ová, G. Warnecke, Error estimates for a combined finite volume-finite element method for nonlinear convection-diffusion problems, SIAM J. Numer. Anal., 36 (1999), 1528-1548. [16] V. Girault and P. -A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithm, Springer Series in Computational Mathematics, Vol 5, Springer-Verlag, Berlin, 1986. [17] F. Guillén-González, P. Damázio, M. A. Rojas-Medar, Approximation by an iterative method for regular solutions for incompressible fluids with mass diffusion, J. Math. Anal. Appl., 326 (2007), 468-487. [18] F. Guillén-González, J. V. Gutiérrez-Santacreu, Unconditional stability and convergence of fully discrete schemes for 2D viscous fluids models with mass diffusion, Mathematics of Computation., 77 (2008), 1495-1524. [19] F. Guillén-González, J. V. Gutiérrez-Santacreu, Conditional stability and convergence of fully discrete scheme for three-dimensional Navier-Stokes equations with mass diffusion, SIAM J. Numer. Anal., 46 (2008), 2276-2308. [20] F. Guillén-González, J. V. Gutiérrez-Santacreu, Error estimates of a linear decoupled Euler-FEM scheme for a mass diffusion model, Numer. Math., 117 (2011), 333-371. [21] A. Kazhikhov, Sh. Smagulov, The correctness of boundary value problems in a diffusion model of an inhomogeneous fluid, Sov. Phys. Dokl., 22 (1977), 249-252. [22] J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969. [23] P. Secchi, On the motion of viscous fluids in the presence of diffusion, SIAM J. Math. Anal., 19 (1988), 22-31. [24] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, 2003. [25] J. Simon, Compact sets in the space $L^p\big(0, T;B\big)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. [26] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Revised Edition, Studies in mathematics and its applications vol. 2, North Holland Publishing Company-Amsterdam, New York, 1984. [27] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamic; A Practical Introduction, Springer-Verlag, Berlin, 2009.

show all references

##### References:
 [1] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and Its Applications, 22, North-Holland, Publishing Co. , Amesterdam, 1990. [2] D. Bresch, E. H. Essoufi, M. Sy, Effects of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397. [3] R. C. Cabrales, F. Guillén-González, J. V. Gutiérrez-Santacreu, Stability and convergence for a complete model of mass diffusion, Applied Numerical Mathematics, 61 (2011), 1161-1185. [4] X. Cai, L. Liao, Y. Sun, Global regularity for the initial value problem of a 2-D Kazhikhov-Smagulov type model, Nonlinear Analysis, 75 (2012), 5975-5983. [5] X. Cai, L. Liao, Y. Sun, Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model, Discrete and Continuous Dynamical Systems, Series S, 7 (2014), 917-923. [6] C. Calgaro, E. Chane-Kane, E. Creusé, T. Goudon, $L^∞$-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios, J. Comput. Physics, 229 (2010), 6027-6046. [7] C. Calgaro, E. Creusé, T. Goudon, An hybrid finite volume-finite element method for variable density incompressible flows, J. Comput. Physics, 227 (2008), 4671-4696. [8] C. Calgaro, E. Creusé, T. Goudon, Modeling and simulation of mixture flows: Application to powder-snow avalanches, Computers and Fluids, 107 (2015), 100-122. [9] C. Calgaro, M. Ezzoug, $L^∞$-stability of IMEX-BDF2 finite volume scheme for convection-diffusion equation, Finite Volumes for Complex Applications Ⅷ -Methods and Theoretical Aspects, 2 (2017), 245-253. [10] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1979. [11] J. Droniou, Finite volume schemes for diffusion equations: Introduction to and review of modern methods, Mathematical Models and Methods in Applied Sciences, 24 (2014), 1575-1619. [12] J. Étienne, P. Saramito, A priori error estimates of the Lagrange-Galerkin method for Kazhikhov-Smagulov type systems, C.R. Acad. Sci. Paris Ser. I, 341 (2005), 769-774. [13] R. Eymard, T. Gallouët, R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, vol. Ⅶ, North-Holland, Amsterdam, (2000), 713-1020. [14] M. Feistauer, J. Felcman, M. Lukáčová-Medvid'ová, On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems, Numerical Methods Partial Differential Equations, 13 (1997), 163-190. [15] M. Feistauer, J. Felcman, M. Lukáčová-Medvid'ová, G. Warnecke, Error estimates for a combined finite volume-finite element method for nonlinear convection-diffusion problems, SIAM J. Numer. Anal., 36 (1999), 1528-1548. [16] V. Girault and P. -A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithm, Springer Series in Computational Mathematics, Vol 5, Springer-Verlag, Berlin, 1986. [17] F. Guillén-González, P. Damázio, M. A. Rojas-Medar, Approximation by an iterative method for regular solutions for incompressible fluids with mass diffusion, J. Math. Anal. Appl., 326 (2007), 468-487. [18] F. Guillén-González, J. V. Gutiérrez-Santacreu, Unconditional stability and convergence of fully discrete schemes for 2D viscous fluids models with mass diffusion, Mathematics of Computation., 77 (2008), 1495-1524. [19] F. Guillén-González, J. V. Gutiérrez-Santacreu, Conditional stability and convergence of fully discrete scheme for three-dimensional Navier-Stokes equations with mass diffusion, SIAM J. Numer. Anal., 46 (2008), 2276-2308. [20] F. Guillén-González, J. V. Gutiérrez-Santacreu, Error estimates of a linear decoupled Euler-FEM scheme for a mass diffusion model, Numer. Math., 117 (2011), 333-371. [21] A. Kazhikhov, Sh. Smagulov, The correctness of boundary value problems in a diffusion model of an inhomogeneous fluid, Sov. Phys. Dokl., 22 (1977), 249-252. [22] J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969. [23] P. Secchi, On the motion of viscous fluids in the presence of diffusion, SIAM J. Math. Anal., 19 (1988), 22-31. [24] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, 2003. [25] J. Simon, Compact sets in the space $L^p\big(0, T;B\big)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. [26] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Revised Edition, Studies in mathematics and its applications vol. 2, North Holland Publishing Company-Amsterdam, New York, 1984. [27] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamic; A Practical Introduction, Springer-Verlag, Berlin, 2009.
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