Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model

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November 2012, 11(6): 2371-2391. doi: 10.3934/cpaa.2012.11.2371

Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model

T. Gallouët ^1, and J.-C. Latché ^2,

1.	Université de Provence, CMI, Marseille, France
2.	Institut de Radioprotection et de Sûret, France

Received October 2010 Revised November 2010 Published April 2012

Abstract
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In this paper, we prove an adaptation of the classical compactness Aubin-Simon lemma to sequences of functions obtained through a sequence of discretizations of a parabolic problem. The main difficulty tackled here is to generalize the classical proof to handle the dependency of the norms controlling each function $u^{(n)}$ of the sequence with respect to $n$. This compactness result is then used to prove the convergence of a numerical scheme combining finite volumes and finite elements for the solution of a reduced turbulence problem.

Keywords: Partial differential equations, convergence, discretization.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: T. Gallouët, J.-C. Latché. Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2371-2391. doi: 10.3934/cpaa.2012.11.2371

References:

[1]	P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J.-L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Annali della Scuola Normale Superiora di Pisa, Classe de Scienze, 22 (1955), 240-273.
[2]	L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis, 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.
[3]	P. G. Ciarlet, "Handbook of Numerical Analysis Volume II: Finite Elements Methods, Basic Error Estimates for Elliptic Problems," Handbook of Numerical Analysis, Volume II, P. Ciarlet and J. L. Lions eds., 1991, 17-351.
[4]	G. Cimatti, Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor, Annali di Matematica Pura ed Applicata, 162 (1992), 33-42. doi: 10.1007/BF01759998.
[5]	S. Clain, "Analyse mathématique et numérique d'un modèle de chauffage par induction," EPFL, 1994.
[6]	M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, Revue Française d'Automatique, Informatique et Recherche Opérationnelle (R.A.I.R.O.), R-3 (1973), 33-75.
[7]	R. Eymard, T . Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis, Volume VII" (P. Ciarlet and J. L. Lions eds), North Holland, (2000), 713-1020.
[8]	R. Eymard, T. Gallouët and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes - SUSHI: a scheme using stabilization and hybrid interfaces, IMA Journal of Numerical Analysis, 30 (2009), 1009-1043. doi: 10.1093/imanum/drn084.
[9]	T. Gallouët, R. Herbin and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: the isothermal case, Mathematics of Computation, 267 (2009), 1333-1352. doi: 10.1090/S0025-5718-09-02216-9.
[10]	T. Gallouët, A. Larcher and J.-C. Latché, Convergence of a finite volume scheme for the convection-diffusion equation with $L^1$ data, Mathematics of Computation, to appear.
[11]	A. Larcher and J.-C. Latché, Convergence analysis of a finite element - finite volume scheme for a RANS turbulence model, submitted.
[12]	R. Lewandowski, The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity, Nonlinear Analysis, Theory, Methods & Applications, 28 (1997), 393-417. doi: 10.1016/0362-546X(95)00149-P.
[13]	J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Paris, 1969.
[14]	R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numerical Methods for Partial Differential Equations, 8 (1992), 97-111. doi: 10.1002/num.1690080202.
[15]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. doi: 10.1007/BF01762360.
[16]	R. Temam, "Navier-Stokes Equations," Studies in mathematics and its applications, North Holland, 1977.

show all references

References:

[1]	P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J.-L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Annali della Scuola Normale Superiora di Pisa, Classe de Scienze, 22 (1955), 240-273.
[2]	L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis, 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.
[3]	P. G. Ciarlet, "Handbook of Numerical Analysis Volume II: Finite Elements Methods, Basic Error Estimates for Elliptic Problems," Handbook of Numerical Analysis, Volume II, P. Ciarlet and J. L. Lions eds., 1991, 17-351.
[4]	G. Cimatti, Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor, Annali di Matematica Pura ed Applicata, 162 (1992), 33-42. doi: 10.1007/BF01759998.
[5]	S. Clain, "Analyse mathématique et numérique d'un modèle de chauffage par induction," EPFL, 1994.
[6]	M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, Revue Française d'Automatique, Informatique et Recherche Opérationnelle (R.A.I.R.O.), R-3 (1973), 33-75.
[7]	R. Eymard, T . Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis, Volume VII" (P. Ciarlet and J. L. Lions eds), North Holland, (2000), 713-1020.
[8]	R. Eymard, T. Gallouët and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes - SUSHI: a scheme using stabilization and hybrid interfaces, IMA Journal of Numerical Analysis, 30 (2009), 1009-1043. doi: 10.1093/imanum/drn084.
[9]	T. Gallouët, R. Herbin and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: the isothermal case, Mathematics of Computation, 267 (2009), 1333-1352. doi: 10.1090/S0025-5718-09-02216-9.
[10]	T. Gallouët, A. Larcher and J.-C. Latché, Convergence of a finite volume scheme for the convection-diffusion equation with $L^1$ data, Mathematics of Computation, to appear.
[11]	A. Larcher and J.-C. Latché, Convergence analysis of a finite element - finite volume scheme for a RANS turbulence model, submitted.
[12]	R. Lewandowski, The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity, Nonlinear Analysis, Theory, Methods & Applications, 28 (1997), 393-417. doi: 10.1016/0362-546X(95)00149-P.
[13]	J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Paris, 1969.
[14]	R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numerical Methods for Partial Differential Equations, 8 (1992), 97-111. doi: 10.1002/num.1690080202.
[15]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. doi: 10.1007/BF01762360.
[16]	R. Temam, "Navier-Stokes Equations," Studies in mathematics and its applications, North Holland, 1977.

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