• 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 and 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. BreschE. H. Essoufi and M. Sy, Effects of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397. 

[3]

R. C. CabralesF. Guillén-González and J. V. Gutiérrez-Santacreu, Stability and convergence for a complete model of mass diffusion, Applied Numerical Mathematics, 61 (2011), 1161-1185. 

[4]

X. CaiL. 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. 

[5]

X. CaiL. 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. 

[6]

C. CalgaroE. Chane-KaneE. 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. 

[7]

C. CalgaroE. Creusé and T. Goudon, An hybrid finite volume-finite element method for variable density incompressible flows, J. Comput. Physics, 227 (2008), 4671-4696. 

[8]

C. CalgaroE. Creusé and T. Goudon, Modeling and simulation of mixture flows: Application to powder-snow avalanches, Computers and Fluids, 107 (2015), 100-122. 

[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. 

[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 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. 

[13]

R. EymardT. Gallouët and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, vol. Ⅶ, North-Holland, Amsterdam, (2000), 713-1020. 

[14]

M. FeistauerJ. 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. 

[15]

M. FeistauerJ. FelcmanM. 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. 

[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-GonzálezP. 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. 

[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. 

[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. 

[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. 

[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. 

[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. BreschE. H. Essoufi and M. Sy, Effects of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397. 

[3]

R. C. CabralesF. Guillén-González and J. V. Gutiérrez-Santacreu, Stability and convergence for a complete model of mass diffusion, Applied Numerical Mathematics, 61 (2011), 1161-1185. 

[4]

X. CaiL. 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. 

[5]

X. CaiL. 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. 

[6]

C. CalgaroE. Chane-KaneE. 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. 

[7]

C. CalgaroE. Creusé and T. Goudon, An hybrid finite volume-finite element method for variable density incompressible flows, J. Comput. Physics, 227 (2008), 4671-4696. 

[8]

C. CalgaroE. Creusé and T. Goudon, Modeling and simulation of mixture flows: Application to powder-snow avalanches, Computers and Fluids, 107 (2015), 100-122. 

[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. 

[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 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. 

[13]

R. EymardT. Gallouët and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, vol. Ⅶ, North-Holland, Amsterdam, (2000), 713-1020. 

[14]

M. FeistauerJ. 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. 

[15]

M. FeistauerJ. FelcmanM. 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. 

[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-GonzálezP. 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. 

[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. 

[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. 

[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. 

[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. 

[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.

[1]

Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917

[2]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

[3]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

[4]

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

[5]

Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339

[6]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[7]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks and Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401

[8]

Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226

[9]

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks and Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195

[10]

Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 387-402. doi: 10.3934/dcdsb.2018109

[11]

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

[12]

Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems and Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947

[13]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[14]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387

[15]

So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

[16]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[17]

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

[18]

Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051

[19]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure and Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[20]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (140)
  • HTML views (176)
  • Cited by (1)

[Back to Top]