六阶椭圆型方程的两类$n$ -矩形有限元
Two families of $n$-rectangle nonconforming finite elements for sixth-order elliptic equations
摘要：
本文提出了求解六阶椭圆型方程的$n$ -任意维矩形网格上的两类离散有限元。建立了新的有限元空间的唯一解性和逼近能力。一个新的机制，称为交换的子矩形，调查拟议的元素的弱连续性被发现。借助于$H^3$问题的某些相容关系，我们建立了三调和方程在任意维数的破$H^3$范数下的拟最优误差估计.通过二维和三维的数值试验进一步验证了理论分析的正确性。
英文摘要：
In this paper, we propose two families of nonconforming finite elements on $n$-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the new finite element spaces are established. A new mechanism, called the exchange of sub-rectangles, for investigating the weak continuities of the proposed elements is discovered. With the help of some conforming relatives for the $H^3$ problems, we establish the quasi-optimal error estimate for the tri-harmonic equation in the broken $H^3$ norm of any dimension. The theoretical results are validated further by the numerical tests in both 2D and 3D situations.
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