THE BEST L2 NORM ERROR ESTIMATE OF LOWER ORDER FINITE ELEMENT METHODS FOR THE FOURTH ORDER PROBLEM
THE BEST L2 NORM ERROR ESTIMATE OF LOWER ORDER FINITE ELEMENT METHODS FOR THE FOURTH ORDER PROBLEM
复制标题
四阶问题低阶有限元方法的最佳L2范数误差估计
DOI：
10.4208/jcm.1203-m3855
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发表时间：
2012-09-01
影响因子：
0.9
通讯作者：
Shi, Zhong-Ci
中科院分区：
文献类型：
--
作者：
Hu, Jun;Shi, Zhong-Ci
本文分析了四阶问题低阶有限元方法的L-2模误差估计。我们证明了有限元解的L-2模的最佳误差估计是二阶的，一般不能改进。的主要成分是饱和条件建立这些元素和一个身份的能量范数的有限元解的错误。这一结果适用于文献中大多数常用的低阶有限元方法，包括：Powell-Sabin C-1-P-2宏元、C-M莫利元、C-1-Q（2）宏元、C-M矩形莫利元和C-M不完全双二次元。另外，只要满足饱和条件，该结果实际上也适用于Adini元、Fraeijs de Veubeke元、Wang-Xu元和Wang-Shi-Xu元。这一结果解决了文献中一个长期存在的问题：四阶问题的低阶有限元方法的L-2模误差估计是否比能量模误差估计高两个阶？
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?