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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

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.
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, 22 (1955), 240.

[2]

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data,, Journal of Functional Analysis, 87 (1989), 149. 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, (1991), 17.

[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. 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 Fran\ccaise d'Automatique, R-3 (1973), 33.

[7]

R. Eymard, T . Gallouët and R. Herbin, Finite volume methods,, in, (2000), 713.

[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. 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. 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, ().

[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, 28 (1997), 393. doi: 10.1016/0362-546X(95)00149-P.

[13]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,", Dunod, (1969).

[14]

R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element,, Numerical Methods for Partial Differential Equations, 8 (1992), 97. 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. doi: 10.1007/BF01762360.

[16]

R. Temam, "Navier-Stokes Equations,", Studies in mathematics and its applications, (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, 22 (1955), 240.

[2]

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data,, Journal of Functional Analysis, 87 (1989), 149. 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, (1991), 17.

[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. 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 Fran\ccaise d'Automatique, R-3 (1973), 33.

[7]

R. Eymard, T . Gallouët and R. Herbin, Finite volume methods,, in, (2000), 713.

[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. 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. 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, ().

[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, 28 (1997), 393. doi: 10.1016/0362-546X(95)00149-P.

[13]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,", Dunod, (1969).

[14]

R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element,, Numerical Methods for Partial Differential Equations, 8 (1992), 97. 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. doi: 10.1007/BF01762360.

[16]

R. Temam, "Navier-Stokes Equations,", Studies in mathematics and its applications, (1977).

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