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Flow invariance for nonautonomous nonlinear partial differential delay equations
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 |
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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