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The F-E-M test for convergence of nonconforming finite elements


Author: Zhong Ci Shi
Journal: Math. Comp. 49 (1987), 391-405
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1987-0906178-7
MathSciNet review: 906178
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Abstract | References | Similar Articles | Additional Information

Abstract: A new convergence test, the F-E-M-Test, is established for the method of nonconforming finite elements. The F-E-M-Test is simple to apply, it checks only the local properties of shape functions along each interface or on each element. The test is valid for a wide class of nonconforming elements in practical applications.


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  • [1] G. P. Bazeley, Y. K. Cheung, B. M. Irons & O. C. Zienkiewicz, Triangular Element in Plate Bending. Conforming and Nonconforming Elements, Proc. 1st Conf. Matrix Methods in Structural Mechanics, AFFDL-TR-CC-80, Wright-Patterson Air Force Base, Ohio, 1965, pp. 547-576.
  • [2] B. Fraeijs de Veubeke, "Variational principles and the patch test," Internat. J. Numer. Methods Engrg., v. 8, 1974, pp. 783-801. MR 0375911 (51:12099)
  • [3] N. Herrmann, Bemerkungen zum Patch-Test, Second Workshop Diskretisierungen in der Kontinuumsmechanik, Bad Honnef, 1985.
  • [4] B. M. Irons & A. Razzaque, Experience With the Patch Test for Convergence of Finite Elements, Proc. Symps. on Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Operators (Baltimore 1972) (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 557-587. MR 0423839 (54:11813)
  • [5] G. Sander & P. Beckers, "The influence of the choice of connectors in the finite element method," Internat. J. Numer. Methods Engrg., v. 11, 1977, pp. 1491-1505. MR 0502734 (58:19666)
  • [6] Z. C. Shi, "An explicit analysis of Stummel's patch test examples," Internat. J. Numer. Methods Engrg., v. 20, 1984, pp. 1233-1246. MR 751335 (85g:65096)
  • [7] Z. C. Shi, "On the convergence properties of the quadrilateral elements of Sander and Beckers," Math. Comp., v. 42, 1984, pp. 493-504. MR 736448 (85h:65245)
  • [8] Z. C. Shi, "A convergence condition for the quadrilateral Wilson element," Numer. Math., v. 44, 1984, pp. 349-361. MR 757491 (86d:65151)
  • [9] Z. C. Shi, "The generalized patch test for Zienkiewicz's triangles," J. Comput. Math., v. 2, 1984, 279-286. MR 815422 (86m:73051)
  • [10] Z. C. Shi, "Convergence properties of two nonconforming finite elements," Comput. Methods Appl. Mech. Engrg., v. 48, 1985, pp. 123-139. MR 784479 (87c:73073)
  • [11] Z. C. Shi, On the Convergence of Nonconforming Finite Elements, Proc. Fifth Internat. Conf. on Differential Equations and Differential Geometry, Science Press, Beijing, 1985. MR 824490 (88a:73052)
  • [12] Z. C. Shi, On Stummel's Examples to the Patch Test, University of Frankfurt, 1985. (Preprint.)
  • [13] Z. C. Shi & F. Stummel, Certain Convergence Criteria for Nonconforming Finite Elements, Workshop Diskretisierungen in der Kontinuumsmechanik, Bad Honnef, 1983.
  • [14] J. C. Simo, "A note on the Stummel problem." (To appear.)
  • [15] B. Specht, "Modified shape functions for the three node plate bending element passing the patch test." (To appear.)
  • [16] G. Strang, Variational Crimes in the Finite Element Method, Proc. Sympos. on Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Operators (Baltimore 1972) (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 689-710. MR 0413554 (54:1668)
  • [17] G. Strang & G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR 0443377 (56:1747)
  • [18] F. Stummel, "The generalized patch test," SIAM J. Numer. Anal., v. 16, 1979, pp. 449-471. MR 530481 (80e:65106)
  • [19] F. Stummel, "The limitations of the patch test," Internat. J. Numer. Methods Engrg., v. 15, 1980, pp. 177-188.
  • [20] F. Stummel, "Basic compactness properties of nonconforming and hybrid finite element spaces," RAIRO Anal. Numer., v. 4, 1980, pp. 81-115. MR 566091 (81h:65058)
  • [21] R. L. Taylor, J. C. Simo, O. C. Zienkiewicz & A. H. C. Chan, "The patch test--a condition for assessing FEM convergence," Internat. J. Numer. Methods Engrg., v. 22, 1986, pp. 39-62. MR 831836 (87c:73075)

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DOI: https://doi.org/10.1090/S0025-5718-1987-0906178-7
Article copyright: © Copyright 1987 American Mathematical Society

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