Abstract:A class of nonconforming quadrilateral membrane elements, introduced by Sander and Beckers, is discussed in all details. It is proved that two elements that do not pass the patch test may still yield convergence under a suitable condition on mesh subdivisions, whereas one such element is found to be divergent for all mesh subdivisions. All other finite elements in this class that pass the patch test are convergent. The mathematical analysis provides a clear explanation of the convergence behavior appearing in the numerical examples of Sander and Beckers.
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Bruce M. Irons and Abdur Razzaque, Experience with the patch test for convergence of finite elements, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 557–587. MR 0423839
- G. Sander and P. Beckers, The influence of the choice of connectors in the finite element method, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 316–342. MR 0502734
- Zhong Ci Shi, An explicit analysis of Stummel’s patch test examples, Internat. J. Numer. Methods Engrg. 20 (1984), no. 7, 1233–1246. MR 751335, DOI 10.1002/nme.1620200705 F. Stummel, "The limitations of the patch test," Internat. J. Numer. Methods Engrg, v. 15, 1980, pp. 177-188.
- Friedrich Stummel, The generalized patch test, SIAM J. Numer. Anal. 16 (1979), no. 3, 449–471. MR 530481, DOI 10.1137/0716037
- Friedrich Stummel, Basic compactness properties of nonconforming and hybrid finite element spaces, RAIRO Anal. Numér. 14 (1980), no. 1, 81–115 (English, with French summary). MR 566091
- © Copyright 1984 American Mathematical Society
- Journal: Math. Comp. 42 (1984), 493-504
- MSC: Primary 65N30; Secondary 73K25
- DOI: https://doi.org/10.1090/S0025-5718-1984-0736448-4
- MathSciNet review: 736448