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New Error Estimates of Nonconforming Finite Element Methods for the Poisson Problem with Low Regularity Solution

Published online by Cambridge University Press:  03 June 2015

Youai Li
Youai Li*
Affiliation:
School of Science, Beijing Technology and Business University, Beijing 100048, China
*
*Corresponding author. Email: lya@lsec.cc.ac.cn
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Abstract

In this paper, we revisit a priori error analysis of nonconforming finite element methods for the Poisson problem. Based on some techniques developed in the context of the a posteriori error analysis, under two reasonable assumptions on the nonconforming finite element spaces, we prove that, up to some oscillation terms, the consistency error can be bounded by the approximation error. We check these two assumptions for the most used lower order nonconforming finite element methods. Compared with the classical error analysis of the nonconforming finite element method, the a priori analysis herein only needs the H1 regularity of the exact solution.

Keywords

65N3065N1535J25Error estimatenonconforming finite elementbubble functionPoisson problemlow regularity

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 2 , April 2014, pp. 179 - 190
Copyright
Copyright © Global-Science Press 2014

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References

[1]Ainsworth, M. and Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience [John Wiley &Sons], New York, 2000.Google Scholar
[2]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer Verlag, 2nd Edition, 2002.CrossRefGoogle Scholar
[3]Bernardi, C. and Glrault, V., A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal., 35 (1998), pp. 1893–1916.Google Scholar
[4]Brenner, S., Poincare-Friedrichs inequality for piecewise H1 functions, SIAM J. Numer. Anal., 41 (2003), pp. 306–324.CrossRefGoogle Scholar
[5]Carstensen, C., Quasi-interpolation and a posteriori error analysis in finite element methods, M2NA, 33 (1999), pp. 1187–1202.CrossRefGoogle Scholar
[6]Carstensen, C. and Hu, J., A unifying theory of a posteriori error control for nonconforming finite element methods, Numer. Math., 107 (2007), pp. 473–502.CrossRefGoogle Scholar
[7]Carstensen, C., Hu, J. and Orlando, A., Framework for the a posteriori error analysis of nonconforming finite elements, SIAM J. Numer. Anal., 45 (2007), pp. 68–82.Google Scholar
[8]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, 1978, reprinted as SIAM Classics in Applied Mathematics, 2002.Google Scholar
[9]CléMent, P., Approximation by finite-element functions using local regularization, RAIRO Anal. Numer., 9 (1975), pp. 77–84.Google Scholar
[10]Crouzeix, M. and Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numer., 7 (1973), pp. 33–76.Google Scholar
[11]Douglas, J., Santos, J. E., Sheen, D. and Ye, X., Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems, Math. Models Numer. Anal., 33 (1999), pp.747–770.CrossRefGoogle Scholar
[12]Gudi, T., A new error analysis for discontinous finite element methods for linear problems, Math. Comput., 79 (2010), pp. 2169–2189.Google Scholar
[13]Han, H. D., Nonconforming elements in the mixed finite element method, J. Comput. Math., 2 (1984), pp. 223–233.Google Scholar
[14]Hu, J., Huang, Y. Q. and Lin, Q., The lower bounds for eigenvalues of elliptic operators by nonconforming finite element methods, arXiv:1112.1145v1[Math.NA], 2011.Google Scholar
[15]Hu, J. and Shi, Z. C., Constrained nonconforming quadrilateral rotated Q1-element, J. Comput. Math., 23 (2005), pp. 561–586.Google Scholar
[16]Lin, Q., Tobiska, L. and Zhou, A. H., On the superconvergence of nonconforming low order finite elements applied to the Poisson equation, IMA. J. Numer. Anal., 25 (2005), pp. 160–181.Google Scholar
[17]Mao, S. P. and Shi, Z. C., On the error bounds of nonconforming finite elements, Sci. China Math., 53 (2010), pp. 2917–2926.Google Scholar
[18]Park, C. and Sheen, D., P1-nonconforming quadrilateral finite element methods for second-order elliptic problems, SIAM J. Numer. Anal., 41 (2003), pp. 624–640.Google Scholar
[19]Rannacher, R. and Turek, S., Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differantial Equations, 8 (1992), pp. 97–111.Google Scholar
[20]Scott, L. R. and Zhang, S. Y., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54 (1990), pp. 483–493.CrossRefGoogle Scholar
[21]Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, 1996.Google Scholar