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Nonconforming finite elements for the Stokes problem

Authors: Michel Crouzeix and Richard S. Falk
Journal: Math. Comp. 52 (1989), 437-456
MSC: Primary 65N30; Secondary 76-08, 76D07
MathSciNet review: 958870
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Abstract: A new stability result is obtained for the approximation of the stationary Stokes problem by nonconforming piecewise cubic approximations to the velocities and a discontinuous piecewise quadratic approximation to the pressure. The basic result is that for most reasonable meshes, these elements form a stable pair without the addition of quartic bubble functions (which had previously been added to insure stability).

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  • [1] C. Bernardi & G. Raugel, "Analysis of some finite elements for the Stokes problem," Math. Comp., v. 44, 1985, pp. 71-79. MR 771031 (86b:65119)
  • [2] J. Boland & R. Nicolaides, "Stability of finite elements under divergence constraints," SIAM J. Numer. Anal., v. 20, 1983, pp. 722-731. MR 708453 (85e:65046)
  • [3] F. Brezzi, "On the existence, uniqueness, and the approximation of saddle point problems arising from Lagrangian multipliers," RAIRO Anal. Numér., v. 8, 1974, pp. 129-151. MR 0365287 (51:1540)
  • [4] M. Crouzeix & P.-A. Raviart, "Conforming and non-conforming finite element methods for solving the stationary Stokes equations," RAIRO Anal. Numér., v. 7, 1973, pp. 33-76. MR 0343661 (49:8401)
  • [5] M. Fortin, Calcul Numérique des Écoulements des Fluides de Bingham et des Fluides Newtoniens Incompressibles par la Méthode des Éléments Finis, Thèse Univ. Paris, 1972.
  • [6] M. Fortin, "Old and new finite elements for incompressible flows," Internat. J. Numer. Methods Fluids, v. 1, 1981, pp. 347-364. MR 633812 (83a:76015)
  • [7] M. Fortin, "A non-conforming piecewise quadratic finite element on triangles," Internat. J. Numer. Methods Engrg., v. 19, 1983, pp. 505-520. MR 702056 (84g:76004)
  • [8] V. Girault & P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. MR 851383 (88b:65129)
  • [9] P.-A. Raviart & J. M. Thomas, "Primal hybrid finite element methods for 2nd order elliptic equations," Math. Comp., v. 31, 1977, pp. 391-413. MR 0431752 (55:4747)
  • [10] L. R. Scott & M. Vogelius, "Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials," Math. Modelling and Numer. Anal., v. 19, 1985, pp. 111-143. MR 813691 (87i:65190)
  • [11] R. Stenberg, "Analysis of mixed finite element methods for the Stokes problem: A unified approach," Math. Comp., v. 42, 1984, pp. 9-23. MR 725982 (84k:76014)

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Keywords: Stokes, finite element, nonconforming
Article copyright: © Copyright 1989 American Mathematical Society

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