On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations

Baker, Garth A.; Dougalis, Vassilios A.; Karakashian, Ohannes A.

doi:10.1090/S0025-5718-1982-0669634-0

On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations
HTML articles powered by AMS MathViewer

by Garth A. Baker, Vassilios A. Dougalis and Ohannes A. Karakashian PDF

Math. Comp. 39 (1982), 339-375 Request permission

Abstract:

We consider approximating the solution of the initial and boundary value problem for the Navier-Stokes equations in bounded two- and three-dimensional domains using a nonstandard Galerkin (finite element) method for the space discretization and the third order accurate, three-step backward differentiation method (coupled with extrapolation for the nonlinear terms) for the time stepping. The resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions; all these linear systems have the same matrix. The resulting approximations of the velocity are shown to have optimal rate of convergence in ${L^2}$ under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space.

References

Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
James H. Bramble, Multistep methods for quasilinear parabolic equations, Computational methods in nonlinear mechanics (Proc. Second Internat. Conf., Univ. Texas, Austin, Tex., 1979) North-Holland, Amsterdam-New York, 1980, pp. 177–183. MR 576904
James H. Bramble and Peter H. Sammon, Efficient higher order single step methods for parabolic problems. I, Math. Comp. 35 (1980), no. 151, 655–677. MR 572848, DOI 10.1090/S0025-5718-1980-0572848-X
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
Jim Douglas Jr., Todd Dupont, and Richard E. Ewing, Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. Numer. Anal. 16 (1979), no. 3, 503–522. MR 530483, DOI 10.1137/0716039
Todd Dupont, Graeme Fairweather, and J. Peter Johnson, Three-level Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 11 (1974), 392–410. MR 403259, DOI 10.1137/0711034
Richard S. Falk, An analysis of the finite element method using Lagrange multipliers for the stationary Stokes equations, Math. Comput. 30 (1976), no. 134, 241–249. MR 0403260, DOI 10.1090/S0025-5718-1976-0403260-0
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
P. Jamet and P.-A. Raviart, Numerical solution of the stationary Navier-Stokes equations by finite element methods, Computing methods in applied sciences and engineering (Proc. Internat. Sympos., Versailles, 1973) Lecture Notes in Comput. Sci., Vol. 10, Springer, Berlin, 1974, pp. 193–223. MR 0448951

A Mathematical Study of Two-Dimensional Incompressible Viscous Flows

O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Second English edition, revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu. MR 0254401
J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR 0423815
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 10.1007/BF01396312
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15 (German). MR 341903, DOI 10.1007/BF02995904
J. Nitsche, On Dirichlet problems using subspaces with nearly zero boundary conditions, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 603–627. MR 0426456
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
Miloš Zlámal, Finite element methods for nonlinear parabolic equations, RAIRO Anal. Numér. 11 (1977), no. 1, 93–107, 113 (English, with French summary). MR 502073, DOI 10.1051/m2an/1977110100931