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 ${H^1}$-norm is established.

- 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**, DOI https://doi.org/10.1017/S0308210500010428 - 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**, DOI https://doi.org/10.1090/S0025-5718-1982-0669634-0
M. Crouzeix & V. Thomée, - 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** - Hiroshi Fujita and Tosio Kato,
*On the Navier-Stokes initial value problem. I*, Arch. Rational Mech. Anal.**16**(1964), 269–315. MR**166499**, DOI https://doi.org/10.1007/BF00276188 - 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**, DOI https://doi.org/10.2969/jmsj/02840749 - Hiroshi Fujita and Hiroko Morimoto,
*On fractional powers of the Stokes operator*, Proc. Japan Acad.**46**(1970), 1141–1143. MR**296755** - 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** - 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**, DOI https://doi.org/10.1137/0719018 - 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**, DOI https://doi.org/10.1137/0723049 - 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**, DOI https://doi.org/10.1137/0725032 - 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**, DOI https://doi.org/10.1137/0727022 - Joseph W. Jerome,
*Approximation of nonlinear evolution systems*, Mathematics in Science and Engineering, vol. 164, Academic Press, Inc., Orlando, FL, 1983. MR**690582** - Tosio Kato,
*Perturbation theory for linear operators*, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR**0407617** - Tosio Kato and Hiroshi Fujita,
*On the nonstationary Navier-Stokes system*, Rend. Sem. Mat. Univ. Padova**32**(1962), 243–260. MR**142928** - 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**, DOI https://doi.org/10.1007/BF01396312 - 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** - 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** - 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** - 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** - Reimund Rautmann,
*On optimum regularity of Navier-Stokes solutions at time $t=0$*, Math. Z.**184**(1983), no. 2, 141–149. MR**716267**, DOI https://doi.org/10.1007/BF01252853 - 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** - 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**, DOI https://doi.org/10.1016/0022-0396%2882%2990075-4 - 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** - Vidar Thomée,
*Galerkin finite element methods for parabolic problems*, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR**744045**

*On the Discretization in Time of Semilinear Parabolic Equations with Non-Smooth Initial Data*, Preprint, Université de Rennes, 1985.

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Article copyright:
© Copyright 1989
American Mathematical Society