Analysis of some mixed finite element methods related to reduced integration
Authors:
Claes Johnson and Juhani Pitkäranta
Journal:
Math. Comp. 38 (1982), 375400
MSC:
Primary 65N30
MathSciNet review:
645657
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Abstract: We prove error estimates for the following two mixed finite element methods related to reduced integration: A method for Stokes' problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure, and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress. The main idea of the proof in the case of Stokes' problem is to combine a weak BabuškaBrezzi type stability estimate for the pressure with a superapproximability property for the velocities. A similar technique is used in the case of the plate problem.
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 [4]
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 P. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
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 M. Crouzeix & P.A. Raviart, "Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I," R.A.I.R.O. Sér. Rouge, v. 7, 1973, pp. 3376. MR 0343661 (49:8401)
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 V. Girault, "A combined finite element and MarkerandCell method for solving NavierStokes equations," Numer. Math., v. 26, 1976, pp. 3959. MR 0449179 (56:7484)
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 V. Girault & P.A. Raviart, Finite Element Approximation of the NavierStokes Equations, Lecture Notes in Math., Vol. 749, SpringerVerlag, Berlin, 1979. MR 548867 (83b:65122)
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 R. Glowinski & O. Pironneau, "On numerical methods for the Stokes problem," in Energy Methods in Finite Element Analysis (Glowinski et al., Eds.), Wiley, New York, 1979. MR 537009 (80j:76024)
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 [12]
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 [13]
 R. L. Sani, P. M. Gresho & R. L. Lee, On the Spurious Pressures Generated by Certain GFEM Solutions of the Incompressible NavierStokes Equations, Technical report, Lawrence Livermore Laboratory, Oct. 1979.
 [14]
 K. Singh, "Reduced integration for improved accuracy of finite element approximations," Comput. Methods Appl. Mech. Engrg., v. 14, 1978, pp. 2337. MR 0495022 (58:13790)
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 R. Temam, "Une méthode d'approximation de la solution des equations NavierStokes," Bull. Soc. Math. France, v. 98, 1968, pp. 115152. MR 0237972 (38:6249)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206456572
PII:
S 00255718(1982)06456572
Article copyright:
© Copyright 1982
American Mathematical Society
