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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. Clain, "Analyse mathématique et numérique d'un modèle de chauffage par induction," EPFL, 1994. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[11]

A. Larcher and J.-C. Latché, Convergence analysis of a finite element - finite volume scheme for a RANS turbulence model,, submitted., ().   Google Scholar

[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.  Google Scholar

[13]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Temam, "Navier-Stokes Equations," Studies in mathematics and its applications, North Holland, 1977.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. Clain, "Analyse mathématique et numérique d'un modèle de chauffage par induction," EPFL, 1994. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[11]

A. Larcher and J.-C. Latché, Convergence analysis of a finite element - finite volume scheme for a RANS turbulence model,, submitted., ().   Google Scholar

[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.  Google Scholar

[13]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Temam, "Navier-Stokes Equations," Studies in mathematics and its applications, North Holland, 1977.  Google Scholar

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