Analysis of locally stabilized mixed finite element methods for the Stokes problem
Authors:
Nasserdine Kechkar and David Silvester
Journal:
Math. Comp. 58 (1992), 110
MSC:
Primary 65N15; Secondary 65N30, 76D07, 76M10
MathSciNet review:
1106973
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References 
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Additional Information
Abstract: In this paper, a locally stabilized finite element formulation of the Stokes problem is analyzed. A macroelement condition which is sufficient for the stability of (locally stabilized) mixed methods based on a piecewise constant pressure approximation is introduced. By satisfying this condition, the stability of the quadrilateral, and the triangular element, can be established.
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 [1]
 I. Babuška, Error bounds for finite element method, Numer. Math. 16 (1971), 322333. MR 0288971 (44:6166)
 [2]
 I. Babuška, J. Osborn, and J. Pitkäranta, Analysis of mixed methods using mesh dependent norms, Math. Comp. 35 (1980), 10391062. MR 583486 (81m:65166)
 [3]
 J. Boland and R. A. Nicolaides, Stability of finite elements under divergence constraints, SIAM J. Numer. Anal. 20 (1983), 722731. MR 708453 (85e:65046)
 [4]
 J. H. Bramble and J. E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp. 50 (1988), 117. MR 917816 (89m:65097a)
 [5]
 F. Brezzi and J. Pitkäranta, On the stabilisation of finite element approximations of the Stokes problem, Efficient Solutions of Elliptic Systems (W. Hackbusch, ed.), Notes on Numerical Fluid Mechanics, vol. 10, Vieweg, Braunschweig, 1984, pp. 1119.
 [6]
 J. Douglas, Jr. and J. Wang, An absolutely stabilized finite element method for the Stokes problem, Math. Comp. 52 (1989), 495508. MR 958871 (89j:65069)
 [7]
 V. Girault and P. A. Raviart, Finite element methods for NavierStokes equations: theory and algorithms, SpringerVerlag, Berlin, Heidelberg, 1986. MR 851383 (88b:65129)
 [8]
 M. D. Gunzburger, Finite element methods for viscous incompressible flows, Academic Press, London, 1989. MR 1017032 (91d:76053)
 [9]
 T. J. R. Hughes and L. P. Franca, A new finite element formulation for CFD: VII. The Stokes problem with various wellposed boundary conditions: Symmetric formulations that converge for all velocity/ pressure spaces, Comput. Methods Appl. Mech. Engrg. 65 (1987), 8596. MR 914609 (89j:76015g)
 [10]
 N. Kechkar, Analysis and application of locally stabilised mixed finite element methods, Ph.D. Thesis, University of Manchester Institute of Science and Technology, 1989.
 [11]
 N. Kechkar and D. J. Silvester, The stabilisation of low order mixed finite elements for incompressible flow, Proc. 5th Internat. Sympos. on Numerical Methods in Engineering (R. Gruber et al., eds.), vol. 2, Computational Mechanics Publications, Southampton, 1989, pp. 111116. MR 1052962
 [12]
 J. Pitkäranta and T. Saarinen, A multigrid version of a simple finite element method for the Stokes problem, Math. Comp. 45 (1985), 114. MR 790640 (86h:65168)
 [13]
 R. L. Sani, P. M. Gresho, R. L. Lee, and D. F. Griffiths, The cause and cure (?) of the spurious pressures generated by certain finite element method solutions of the incompressible NavierStokes equations, Parts 1 and 2, Internat. J. Numer. Methods Fluids 1 (1981), 1743; 171204. MR 621064 (83i:65083b)
 [14]
 D. J. Silvester and N. Kechkar, Stabilised bilinearconstant velocitypressure finite elements for the conjugate gradient solution of the Stokes problem, Comput. Methods Appl. Mech. Engrg. 79 (1990), 7186. MR 1044204 (90m:76011)
 [15]
 R. Stenberg, Analysis of mixed finite elements for the Stokes problem: A unified approach, Math. Comp. 42 (1984), 923. MR 725982 (84k:76014)
 [16]
 R. Verfürth, A multilevel algorithm for mixed problems, SIAM J. Numer. Anal. 21 (1984), 264271. MR 736330 (85f:65112)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819921106973X
PII:
S 00255718(1992)1106973X
Keywords:
Stabilized finite element,
Stokes equation
Article copyright:
© Copyright 1992 American Mathematical Society
