The Best $L^2$ Norm Error Estimate of Lower Order Finite Element Methods for the Fourth Order Problem
Year: 2012
Journal of Computational Mathematics, Vol. 30 (2012), Iss. 5 : pp. 449–460
Abstract
In the paper, we analyze the $L^2$ norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the $L^2$ norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including: the Powell-Sabin $C^1-P_2$ macro element, the nonconforming Morley element, the $C^1-Q_2$ macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang-Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature: can the $L^2$ norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm?
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1203-m3855
Journal of Computational Mathematics, Vol. 30 (2012), Iss. 5 : pp. 449–460
Published online: 2012-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 12
Keywords: $L^2$ norm error estimate Energy norm error estimate Conforming Nonconforming The Kirchhoff plate.