## 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**
G. A. Baker, "Galerkin approximations to the Navier-Stokes equations." (To appear.)
- 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**
W. Jureidini, - 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

*A Mathematical Study of Two-Dimensional Incompressible Viscous Flows*, Ph.D. Thesis, Harvard University, Cambridge, Mass., 1980.

## Additional Information

- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp.
**39**(1982), 339-375 - MSC: Primary 65M60; Secondary 65N30, 76D05
- DOI: https://doi.org/10.1090/S0025-5718-1982-0669634-0
- MathSciNet review: 669634