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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 -norm is established.
- [1] Herbert Amann, Existence and stability of solutions for semi-linear parabolic systems, and applications to some diffusion reaction equations, Proc. Roy. Soc. Edinburgh Sect. A 81 (1978), no. 1-2, 35–47. MR 529375, https://doi.org/10.1017/S0308210500010428
- [2] Garth A. Baker, Vassilios A. Dougalis, and Ohannes A. Karakashian, On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations, Math. Comp. 39 (1982), no. 160, 339–375. MR 669634, https://doi.org/10.1090/S0025-5718-1982-0669634-0
- [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. Foiaş and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. (9) 58 (1979), no. 3, 339–368. MR 544257
- [5] Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 166499, https://doi.org/10.1007/BF00276188
- [6] Hiroshi Fujita and Akira Mizutani, On the finite element method for parabolic equations. I. Approximation of holomorphic semi-groups, J. Math. Soc. Japan 28 (1976), no. 4, 749–771. MR 428733, https://doi.org/10.2969/jmsj/02840749
- [7] Hiroshi Fujita and Hiroko Morimoto, On fractional powers of the Stokes operator, Proc. Japan Acad. 46 (1970), 1141–1143. MR 296755
- [8] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
- [9] John G. Heywood and Rolf Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), no. 2, 275–311. MR 650052, https://doi.org/10.1137/0719018
- [10] John G. Heywood and Rolf Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal. 23 (1986), no. 4, 750–777. MR 849281, https://doi.org/10.1137/0723049
- [11] John G. Heywood and Rolf Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. III. Smoothing property and higher order error estimates for spatial discretization, SIAM J. Numer. Anal. 25 (1988), no. 3, 489–512. MR 942204, https://doi.org/10.1137/0725032
- [12] John G. Heywood and Rolf Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), no. 2, 353–384. MR 1043610, https://doi.org/10.1137/0727022
- [13] Joseph W. Jerome, Approximation of nonlinear evolution systems, Mathematics in Science and Engineering, vol. 164, Academic Press, Inc., Orlando, FL, 1983. MR 690582
- [14] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
- [15] Tosio Kato and Hiroshi Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962), 243–260. MR 142928
- [16] Marie-Noëlle Le Roux, Méthodes multipas pour des équations paraboliques non linéaires, Numer. Math. 35 (1980), no. 2, 143–162 (French, with English summary). MR 585243, https://doi.org/10.1007/BF01396312
- [17] Hisashi Okamoto, On the semidiscrete finite element approximation for the nonstationary Stokes equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 1, 241–260. MR 657878
- [18] Hisashi Okamoto, On the semidiscrete finite element approximation for the nonstationary Navier-Stokes equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 613–651. MR 687594
- [19] Rolf Rannacher, Stable finite element solutions to nonlinear parabolic problems of Navier-Stokes type, Computing methods in applied sciences and engineering, V (Versailles, 1981) North-Holland, Amsterdam, 1982, pp. 301–309. MR 784647
- [20] Reimund Rautmann, A semigroup approach to error estimates for nonstationary Navier-Stokes approximations, Approximation and optimization in mathematical physics (Oberwolfach, 1982) Methoden Verfahren Math. Phys., vol. 27, Peter Lang, Frankfurt am Main, 1983, pp. 63–77. MR 763003
- [21] Reimund Rautmann, On optimum regularity of Navier-Stokes solutions at time 𝑡=0, Math. Z. 184 (1983), no. 2, 141–149. MR 716267, https://doi.org/10.1007/BF01252853
- [22] Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444
- [23] R. Temam, Behaviour at time 𝑡=0 of the solutions of semilinear evolution equations, J. Differential Equations 43 (1982), no. 1, 73–92. MR 645638, https://doi.org/10.1016/0022-0396(82)90075-4
- [24] Roger Temam, Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. MR 764933
- [25] Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045
Retrieve articles in Mathematics of Computation with MSC: 65M10, 35Q10, 76-08, 76D05
Retrieve articles in all journals with MSC: 65M10, 35Q10, 76-08, 76D05
Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1989-0969488-5
Article copyright:
© Copyright 1989
American Mathematical Society