A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem
A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem
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DOI：
10.1007/s10092-011-0047-8
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发表时间：
2012-06
期刊：
影响因子：
1.7
通讯作者：
J. Guzmán;D. Leykekhman;M. Neilan
中科院分区：
文献类型：
--
作者：
J. Guzmán;D. Leykekhman;M. Neilan
本文讨论了一类小参数ε的四阶奇摄动问题的几个问题。首先，我们引入了一个新的不符合元素族。然后证明了相应的有限元方法对参数ε具有鲁棒性，并且一致收敛于阶h /2。此外，我们还分析了用Nitsche方法弱处理Neumann边界条件的效果。我们发现，当参数ε小于网格尺寸时，这种处理方法更有效，并且可以获得更清晰的误差估计。这种误差分析并不局限于所提出的单元，只要施加弱的诺伊曼边界条件，就可以很容易地对其他单元进行误差分析。最后，讨论了区域内部边界层的局部误差估计和污染效应。
In this paper we address several issues arising from a singularly perturbed fourth order problem with small parameterε. First, we introduce a new family of non-conforming elements. We then prove that the corresponding finite element method is robust with respect to the parameterεand uniformly convergent to orderh1/2. In addition, we analyze the effect of treating the Neumann boundary condition weakly by Nitsche’s method. We show that such treatment is superior when the parameterεis smaller than the mesh sizehand obtain sharper error estimates. Such error analysis is not restricted to the proposed elements and can easily be carried out to other elements as long as the Neumann boundary condition is imposed weakly. Finally, we discuss the local error estimates and the pollution effect of the boundary layers in the interior of the domain.