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.
翻译:在本文中,我们建议用任何维度的美元对角线的两组不兼容的有限元素来解决第六级椭圆方程。 新的有限元素空间的单溶性属性和近似能力已经建立。 发现了一个新的机制,称为子矩形交换机制,用于调查拟议元素的微弱关联性。 在一些符合要求的亲属的帮助下,我们为任何维度的损坏的3- 3美元标准中的3- 和谐方程确定了准最佳误差估计值。 在2D 和 3D 两种情况中,理论结果得到了进一步验证。</s>