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：本文提出两类基于任意维度$n$-矩形网格的非协调有限元，用于求解六阶椭圆问题。建立了新有限元空间的唯一可解性与逼近能力，并发现一种名为"子矩形交换"的新机制，用以研究所提单元的弱连续性。借助$H^3$问题的一些协调元，我们建立了任意维度下三调和方程在分片$H^3$范数中的拟最优误差估计。二维与三维数值实验进一步验证了理论结果。