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On the convergence of a time discretization scheme for the Navier-Stokes equations


Author: T. Geveci
Journal: Math. Comp. 53 (1989), 43-53
MSC: Primary 65M10; Secondary 35Q10, 76-08, 76D05
DOI: https://doi.org/10.1090/S0025-5718-1989-0969488-5
MathSciNet review: 969488
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Abstract: A linearized version of the implicit Euler scheme is considered for the approximation of the solutions to the Navier-Stokes equations in a two-dimensional domain. The rate of convergence in the $ {H^1}$-norm is established.


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  • [1] H. Amann, "Existence and stability of solutions for semilinear parabolic systems and applications to some diffusion reaction equations," Proc. Roy. Soc. Edinburgh Sect. A, v. 81, 1978, pp. 35-47. MR 529375 (80b:35078)
  • [2] G. A. Baker, V. A. Dougalis ic O. A. Karakashian, "On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations," Math. Comp., v. 39, 1982, pp. 339-375. MR 669634 (84h:65096)
  • [3] M. Crouzeix & V. Thomée, On the Discretization in Time of Semilinear Parabolic Equations with Non-Smooth Initial Data, Preprint, Université de Rennes, 1985.
  • [4] C. Foias & R. Temam, "Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations," J. Math. Pures Appl., v. 58, 1979, pp. 339-368. MR 544257 (81k:35130)
  • [5] H. Fujita & T. Kato, "On the Navier-Stokes initial value problem. I," Arch. Rational Mech. Anal., v. 16, 1964, pp. 269-315. MR 0166499 (29:3774)
  • [6] H. Fujita & A. Mizutani, "On the finite element method for parabolic equations, I: Approximation of holomorphic semi-groups," J. Math. Soc. Japan, v. 28, 1976, pp. 749-771. MR 0428733 (55:1753)
  • [7] H. Fujita & H. Morimoto, "On fractional powers of the Stokes operator," Proc. Japan Acad., v. 46, 1970, pp. 1141-1143. MR 0296755 (45:5814)
  • [8] V. Girault & P. A. Raviart, Finite Element Approximation of Navier-Stokes Equations, Lecture Notes in Math., vol. 749, Springer-Verlag, Berlin and New York, 1979. MR 548867 (83b:65122)
  • [9] J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order estimates for spatial discretization," SIAM J. Numer. Anal., v. 19, 1982, pp. 275-311. MR 650052 (83d:65260)
  • [10] J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary Navier-Stokes problem, Part II: Stability of solutions and error estimates uniform in time," SIAM J. Numer. Anal., v. 23, 1986, pp. 750-777. MR 849281 (88b:65132)
  • [11] J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary Navier-Stokes problem, Part III. Smoothing property and higher order error estimates for spatial discretization," SIAM J. Numer. Anal., v. 25, 1988, pp. 489-512. MR 942204 (89k:65114)
  • [12] J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second order time discretization," Preprint. MR 1043610 (92c:65133)
  • [13] J. W. Jerome, Approximation of Nonlinear Evolution Systems, Academic Press, New York and London, 1983. MR 690582 (85g:35064)
  • [14] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin and New York, 1976. MR 0407617 (53:11389)
  • [15] T. Kato & H. Fujita, "On the non-stationary Navier-Stokes system," Rend. Sem. Mat. Univ. Padova, v. 32, 1962, pp. 243-260. MR 0142928 (26:495)
  • [16] M. N. Le Roux, "Méthodes multipas pour des équations paraboliques non linéaires," Numer. Math., v. 35, 1980, pp. 143-162. MR 585243 (81i:65075)
  • [17] H. Okamoto, "On the semi-discrete finite element approximation for the nonstationary Stokes equation," J. Fac. Sci. Univ. Tokyo Sect. IA, v. 29, 1982, pp. 241-260. MR 657878 (83h:65111)
  • [18] H. Okamoto, "On the semi-discrete finite element approximation for the nonstationary Navier-Stokes equation," J. Fac. Sci. Univ. Tokyo Sect. IA, v. 29, 1982, pp. 613-651. MR 687594 (84e:65115)
  • [19] R. Rannacher, "Stable finite element solutions to nonlinear parabolic problems of Navier-Stokes type," in Computing Methods in Applied Sciences and Engineering V (R. Glowinski and J. L. Lions, eds.), North-Holland, Amsterdam, 1982, pp. 301-309. MR 784647 (86h:65145)
  • [20] R. Rautmann, "A semigroup approach to error estimates for nonstationary Navier-Stokes approximations," Methoden Verfahren Math. Phys., v. 27, 1983, pp. 63-77. MR 763003 (86b:65127)
  • [21] R. Rautmann, "On optimum regularity of Navier-Stokes solutions at time $ t = 0$," Math. Z., v. 184, 1983, pp. 141-149. MR 716267 (86a:35118)
  • [22] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 2nd ed., North-Holland, Amsterdam, 1979. MR 603444 (82b:35133)
  • [23] R. Temam, "Behaviour at time $ t = 0$ of the solutions of semilinear evolution equations," J. Differential Equations, v. 43, 1982, pp. 73-92. MR 645638 (83c:35058)
  • [24] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conf. Ser. in Applied Math., vol. 41, SIAM, Philadelphia, 1983. MR 764933 (86f:35152)
  • [25] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Math., vol. 1054, Springer-Verlag, Berlin and New York, 1984. MR 744045 (86k:65006)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1989-0969488-5
Article copyright: © Copyright 1989 American Mathematical Society

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