# American Institute of Mathematical Sciences

July  2012, 17(5): 1383-1405. doi: 10.3934/dcdsb.2012.17.1383

## Convergence results for the vector penalty-projection and two-step artificial compressibility methods

 1 Aix-Marseille Université, Laboratoire d’Analyse, Topologie, Probabilités - CNRS UMR7353, Centre de Mathématiques et Informatique, 13453 Marseille cedex 13, France 2 Université de Bordeaux & IPB, Institut Mathématiques de Bordeaux - CNRS UMR5251, ENSEIRB-MATMECA, Talence, France

Received  July 2011 Revised  December 2011 Published  March 2012

In this paper, we propose and analyze a new artificial compressibility splitting method which is issued from the recent vector penalty-projection method for the numerical solution of unsteady incompressible viscous flows introduced in [1], [2] and [3]. This method may be viewed as an hybrid two-step prediction-correction method combining an artificial compressibility method and an augmented Lagrangian method without inner iteration. The perturbed system can be viewed as a new approximation to the incompressible Navier-Stokes equations. In the main result, we establish the convergence of solutions to the weak solutions of the Navier-Stokes equations when the penalty parameter tends to zero.
Citation: Philippe Angot, Pierre Fabrie. Convergence results for the vector penalty-projection and two-step artificial compressibility methods. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1383-1405. doi: 10.3934/dcdsb.2012.17.1383
##### References:
 [1] Ph. Angot, J.-P. Caltagirone and P. Fabrie, Vector penalty-projection method for the solution of unsteady incompressible flows, in "Finite Volumes for Complex Applications V'' (eds. R. Eymard and J.-M. Hérard), ISTE, London, (2008), 169-176. [2] Ph. Angot, J.-P. Caltagirone and P. Fabrie, A spectacular vector penalty-projection method for Darcy and Navier-Stokes problems, in "Finite Volumes for Complex Applications VI'' (eds J. Fořt, et al.), International Symposium FVCA6 in Prague, June 6-10, Springer Proceedings in Mathematics, 4, Vol. 1, Springer-Verlag, Berlin, (2011), 39-47. [3] Ph. Angot, J.-P. Caltagirone and P. Fabrie, A new fast method to compute saddle-points in constrained optimization and applications, Applied Mathematics Letters, 25 (2012), 245-251. doi: 10.1016/j.aml.2011.08.015. [4] Ph. Angot, J.-P. Caltagirone and P. Fabrie, A fast vector penalty-projection method for in-compressible non-homogeneous or multiphase Navier-stokes problems, Applied Mathematics Letters, 2012, in press. doi: 10.1016/j.aml.2012.01.037. [5] F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications, 52, Springer-Verlag, 2006. [6] A. J. Chorin, A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), 12-26. doi: 10.1016/0021-9991(67)90037-X. [7] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2. [8] C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Annali della Scuolo Normale Superiore di Pisa, Classe di Scienze (4), 5 (1978), 28-63. [9] V. Girault and P.-A. Raviart, "Finite Element Methods for the Navier-Stokes Equations. Theory and Algorithms," Springer Series in Comput. Math., 5, Springer-Verlag, Berlin, 1986. [10] J.-L. Guermond, P. D. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 6011-6045. doi: 10.1016/j.cma.2005.10.010. [11] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd edition, Mathematics and its Applications, Vol. 2 , Gordon and Breach, Science Publishers, New York-London-Paris, 1969. [12] J. Leray, Essai sur les mouvements plans d'un liquide visqueux que limitent des parois, J. Math. Pures Appl., 13 (1934), 331-418. [13] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod & Gauthier-Villars, Paris, 1969. [14] J. Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J. Numer. Anal., 32 (1995), 386-403. doi: 10.1137/0732016. [15] J. Shen, On a new pseudocompressibility method for the incompressible Navier-Stokes equations, Appl. Numer. Math., 21 (1996), 71-90. doi: 10.1016/0168-9274(95)00132-8. [16] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. [17] R. Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France, 96 (1968), 115-152. [18] R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. I, Arch. Ration. Mech. Anal., 32 (1969), 135-153. [19] R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II, Arch. Ration. Mech. Anal., 33 (1969), 377-385. [20] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.

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##### References:
 [1] Ph. Angot, J.-P. Caltagirone and P. Fabrie, Vector penalty-projection method for the solution of unsteady incompressible flows, in "Finite Volumes for Complex Applications V'' (eds. R. Eymard and J.-M. Hérard), ISTE, London, (2008), 169-176. [2] Ph. Angot, J.-P. Caltagirone and P. Fabrie, A spectacular vector penalty-projection method for Darcy and Navier-Stokes problems, in "Finite Volumes for Complex Applications VI'' (eds J. Fořt, et al.), International Symposium FVCA6 in Prague, June 6-10, Springer Proceedings in Mathematics, 4, Vol. 1, Springer-Verlag, Berlin, (2011), 39-47. [3] Ph. Angot, J.-P. Caltagirone and P. Fabrie, A new fast method to compute saddle-points in constrained optimization and applications, Applied Mathematics Letters, 25 (2012), 245-251. doi: 10.1016/j.aml.2011.08.015. [4] Ph. Angot, J.-P. Caltagirone and P. Fabrie, A fast vector penalty-projection method for in-compressible non-homogeneous or multiphase Navier-stokes problems, Applied Mathematics Letters, 2012, in press. doi: 10.1016/j.aml.2012.01.037. [5] F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications, 52, Springer-Verlag, 2006. [6] A. J. Chorin, A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), 12-26. doi: 10.1016/0021-9991(67)90037-X. [7] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2. [8] C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Annali della Scuolo Normale Superiore di Pisa, Classe di Scienze (4), 5 (1978), 28-63. [9] V. Girault and P.-A. Raviart, "Finite Element Methods for the Navier-Stokes Equations. Theory and Algorithms," Springer Series in Comput. Math., 5, Springer-Verlag, Berlin, 1986. [10] J.-L. Guermond, P. D. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 6011-6045. doi: 10.1016/j.cma.2005.10.010. [11] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd edition, Mathematics and its Applications, Vol. 2 , Gordon and Breach, Science Publishers, New York-London-Paris, 1969. [12] J. Leray, Essai sur les mouvements plans d'un liquide visqueux que limitent des parois, J. Math. Pures Appl., 13 (1934), 331-418. [13] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod & Gauthier-Villars, Paris, 1969. [14] J. Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J. Numer. Anal., 32 (1995), 386-403. doi: 10.1137/0732016. [15] J. Shen, On a new pseudocompressibility method for the incompressible Navier-Stokes equations, Appl. Numer. Math., 21 (1996), 71-90. doi: 10.1016/0168-9274(95)00132-8. [16] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. [17] R. Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France, 96 (1968), 115-152. [18] R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. I, Arch. Ration. Mech. Anal., 32 (1969), 135-153. [19] R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II, Arch. Ration. Mech. Anal., 33 (1969), 377-385. [20] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.
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