Analysis of mixed finite elements methods for the Stokes problem: a unified approach
Author:
Rolf Stenberg
Journal:
Math. Comp. 42 (1984), 923
MSC:
Primary 7608; Secondary 76D07
MathSciNet review:
725982
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Abstract: We develop a method for the analysis of mixed finite element methods for the Stokes problem in the velocitypressure formulation. A technical "macroelement condition", which is sufficient for the classical BabuškaBrezzi inequality to be valid, is introduced. Using this condition,we are able to verify the stability, and optimal order of convergence, of several known mixed finite element methods.
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R. Verfürt, Error Estimates for a Mixed Finite Element Approximation of the Stokes Equations, RuhrUniversität Bochum, 1982. (Preprint.)
 [1]
 I. Babuška, "The finite element method with Lagrangian multipliers," Numer. Math., v. 20, 1973, pp. 179192. MR 0359352 (50:11806)
 [2]
 M. Bercovier, "Perturbation of mixed variational problems. Application to mixed finite element methods", RAIRO Anal. Numer., v. 12, 1978, pp. 211236. MR 509973 (80b:49031)
 [3]
 M. Bercovier & M. Engelman, "A finite element method for the numerical solution of viscous incompressible flows". J. Comput. Phys., v. 30, 1979, pp. 181201. MR 528199 (80c:65200)
 [4]
 M. Bercovier & O. Pironneau, "Error estimates for finite element solution of the Stokes problem in the primitive variables," Numer. Math., v. 33, 1979, pp. 211224. MR 549450 (81g:65145)
 [5]
 F. Brezzi, "On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers." RAIRO Ser. Rouge, v. 8, 1974, pp. 129151. MR 0365287 (51:1540)
 [6]
 P. G. Ciarlet, The Finite Element Method fur Elliptic Problems, NorthHolland, Amsterdam. 1978. MR 0520174 (58:25001)
 [7]
 P. G. Ciarlet & P. A. Raviart, "Interpolation theory over curved elements, with applications to finite element methods," Comput. Methods Appl. Mech. Engrg., v. 1, 1972, pp. 217249. MR 0375801 (51:11991)
 [8]
 P. Clement, "Approximation by finite elements using local regularization." RAIRO Ser. Rouge, v. 9, 1975, pp. 7784. MR 0400739 (53:4569)
 [9]
 M. Crouzeix & P. A. Raviart, "Conforming and nonconforming finite element methods for solving the stationary Stokes equations," RAIRO Ser. Rouge, v. 7, 1973, pp. 3376. MR 0343661 (49:8401)
 [10]
 M. Engelman, R. Sani, P. Gresho & M Bercovier, "Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements," Internat. J. Numer. Methods Fluids, v. 2, 1982, pp. 2542. MR 643172 (83a:76006)
 [11]
 V. Girault & P. A. Raviart, Finite Element Approximation of the NavierStokes Equations, Lecture Notes in Math., Vol. 749. Springer, Berlin. 1979. MR 548867 (83b:65122)
 [12]
 P. Hood & C. Taylor, "NavierStokes equations using mixed interpolation," Finite Element Methods in Flow Problems (J. T. Oden. ed.), UAH Press, Huntsville, Alabama, 1974, pp. 121131.
 [13]
 T. J. Hughes, W. K. Liu & A. Brooks, "Finite element analysis of incompressible viscous flows by the penalty function formulation," J. Comput. Phys., v. 30, 1979, pp. 160. MR 524162 (80b:76008)
 [14]
 P. Huyakorn. C. Taylor, R. Lee & P. Gresho, "A comparison of various mixedinterpolation finite elements for the NavierStokes equations," Comput. & Fluids, v. 6, 1978, pp. 2535.
 [15]
 C. Johnson & J. Pitkäranta, "Analysis of some mixed finite element methods related to reduced integration." Math. Comp., v. 38, 1982, pp. 375400. MR 645657 (83d:65287)
 [16]
 P. Le Tallec, "Compatibility condition and existence results in discrete finite incompressible elasticity," Comput. Methods Appl. Mech. Engrg., v. 27, 1981, pp. 239259. MR 629739 (82j:73036)
 [17]
 D. Malkus & T. J. Hughes, "Mixed finite element methodsreduced and selective integration techniques: a unification of concepts," Comput. Mehtods Appl. Mech. Engrg., v. 15, 1978, pp. 6381.
 [18]
 L. Mansfield, "Finite element subspaces with optimal rates of convergence for the stationary Stokes problem." RAIRO Anal. Numer., v. 16, 1982, pp. 4966. MR 648745 (83d:65294)
 [19]
 J. Pitkäranta, "On a mixed finite element method for the Stokes problem in . RAIRO Anal. Numer., v. 16, 1982, pp. 275291. MR 672419 (84i:76031)
 [20]
 J. Pitkäranta & R. Stenberg, "Analysis of some mixed finite element methods for plane elasticity equations," Math. Comp., v. 41, 1983, pp. 399423. MR 717693 (85b:65099)
 [21]
 R. Sani, P. Gresho, R. Lee & D. Griffiths, "The cause and cure (?) of the spurious pressures generated by certain GFEM solutions of the incompressible NavierStokes equations," Internat. J. Numer. Methods Fluids, v. 1, 1981, Part 1, pp. 1744; Part 2, pp. 171204. MR 621064 (83i:65083b)
 [22]
 R. Stenberg, Mixed Finite Element Methods for Two Problems in Elasticity Theory and Fluid Mechanics, Licentiate thesis, Helsinki University of Technology, 1981.
 [23]
 R. Verfürt, Error Estimates for a Mixed Finite Element Approximation of the Stokes Equations, RuhrUniversität Bochum, 1982. (Preprint.)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407259829
PII:
S 00255718(1984)07259829
Article copyright:
© Copyright 1984 American Mathematical Society
