# American Institute of Mathematical Sciences

• Previous Article
On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation
• CPAA Home
• This Issue
• Next Article
The regularity of some vector-valued variational inequalities with gradient constraints
March  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.  Google Scholar [2] D. Bresch, E. H. Essoufi and M. Sy, Effects of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397.   Google Scholar [3] R. C. Cabrales, F. Guillén-González and J. V. Gutiérrez-Santacreu, Stability and convergence for a complete model of mass diffusion, Applied Numerical Mathematics, 61 (2011), 1161-1185.   Google Scholar [4] X. Cai, L. Liao and Y. Sun, Global regularity for the initial value problem of a 2-D Kazhikhov-Smagulov type model, Nonlinear Analysis, 75 (2012), 5975-5983.   Google Scholar [5] X. Cai, L. Liao and 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.   Google Scholar [6] C. Calgaro, E. Chane-Kane, E. Creusé and 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.   Google Scholar [7] C. Calgaro, E. Creusé and T. Goudon, An hybrid finite volume-finite element method for variable density incompressible flows, J. Comput. Physics, 227 (2008), 4671-4696.   Google Scholar [8] C. Calgaro, E. Creusé and T. Goudon, Modeling and simulation of mixture flows: Application to powder-snow avalanches, Computers and Fluids, 107 (2015), 100-122.   Google Scholar [9] C. Calgaro and 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.   Google Scholar [10] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1979.  Google Scholar [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.   Google Scholar [12] J. Étienne and 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.   Google Scholar [13] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, vol. Ⅶ, North-Holland, Amsterdam, (2000), 713-1020.   Google Scholar [14] M. Feistauer, J. Felcman and 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.   Google Scholar [15] M. Feistauer, J. Felcman, M. Lukáčová-Medvid'ová and 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.   Google Scholar [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.  Google Scholar [17] F. Guillén-González, P. Damázio and 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.   Google Scholar [18] F. Guillén-González and 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.   Google Scholar [19] F. Guillén-González and 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.   Google Scholar [20] F. Guillén-González and J. V. Gutiérrez-Santacreu, Error estimates of a linear decoupled Euler-FEM scheme for a mass diffusion model, Numer. Math., 117 (2011), 333-371.   Google Scholar [21] A. Kazhikhov and Sh. Smagulov, The correctness of boundary value problems in a diffusion model of an inhomogeneous fluid, Sov. Phys. Dokl., 22 (1977), 249-252.   Google Scholar [22] J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [23] P. Secchi, On the motion of viscous fluids in the presence of diffusion, SIAM J. Math. Anal., 19 (1988), 22-31.   Google Scholar [24] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, 2003.  Google Scholar [25] J. Simon, Compact sets in the space $L^p\big(0, T;B\big)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.   Google Scholar [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.  Google Scholar [27] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamic; A Practical Introduction, Springer-Verlag, Berlin, 2009.  Google Scholar

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.  Google Scholar [2] D. Bresch, E. H. Essoufi and M. Sy, Effects of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397.   Google Scholar [3] R. C. Cabrales, F. Guillén-González and J. V. Gutiérrez-Santacreu, Stability and convergence for a complete model of mass diffusion, Applied Numerical Mathematics, 61 (2011), 1161-1185.   Google Scholar [4] X. Cai, L. Liao and Y. Sun, Global regularity for the initial value problem of a 2-D Kazhikhov-Smagulov type model, Nonlinear Analysis, 75 (2012), 5975-5983.   Google Scholar [5] X. Cai, L. Liao and 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.   Google Scholar [6] C. Calgaro, E. Chane-Kane, E. Creusé and 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.   Google Scholar [7] C. Calgaro, E. Creusé and T. Goudon, An hybrid finite volume-finite element method for variable density incompressible flows, J. Comput. Physics, 227 (2008), 4671-4696.   Google Scholar [8] C. Calgaro, E. Creusé and T. Goudon, Modeling and simulation of mixture flows: Application to powder-snow avalanches, Computers and Fluids, 107 (2015), 100-122.   Google Scholar [9] C. Calgaro and 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.   Google Scholar [10] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1979.  Google Scholar [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.   Google Scholar [12] J. Étienne and 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.   Google Scholar [13] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, vol. Ⅶ, North-Holland, Amsterdam, (2000), 713-1020.   Google Scholar [14] M. Feistauer, J. Felcman and 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.   Google Scholar [15] M. Feistauer, J. Felcman, M. Lukáčová-Medvid'ová and 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.   Google Scholar [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.  Google Scholar [17] F. Guillén-González, P. Damázio and 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.   Google Scholar [18] F. Guillén-González and 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.   Google Scholar [19] F. Guillén-González and 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.   Google Scholar [20] F. Guillén-González and J. V. Gutiérrez-Santacreu, Error estimates of a linear decoupled Euler-FEM scheme for a mass diffusion model, Numer. Math., 117 (2011), 333-371.   Google Scholar [21] A. Kazhikhov and Sh. Smagulov, The correctness of boundary value problems in a diffusion model of an inhomogeneous fluid, Sov. Phys. Dokl., 22 (1977), 249-252.   Google Scholar [22] J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [23] P. Secchi, On the motion of viscous fluids in the presence of diffusion, SIAM J. Math. Anal., 19 (1988), 22-31.   Google Scholar [24] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, 2003.  Google Scholar [25] J. Simon, Compact sets in the space $L^p\big(0, T;B\big)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.   Google Scholar [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.  Google Scholar [27] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamic; A Practical Introduction, Springer-Verlag, Berlin, 2009.  Google Scholar
 [1] Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 [2] Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 [3] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [4] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [5] Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 [6] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [7] Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127 [8] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351 [9] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 [10] Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 [11] Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 [12] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [13] Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020033 [14] Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006 [15] Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143 [16] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108 [17] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 [18] Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180 [19] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [20] Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029

2019 Impact Factor: 1.105