Two Families of $n$-Rectangle Nonconforming Finite Elements for Sixth-Order Elliptic Equations (Journal of Computational Mathematics, 2024)
In this paper, we propose two families of nonconforming finite elements on $n$-rectanglemeshes of any dimension to solve the sixth-order elliptic equations. The unisolvent propertyand the approximation ability of the new finite element spaces are established. A newmechanism, called the exchange of sub-rectangles, for investigating the weak continuitiesof 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 triharmonic equationin the broken $H^3$ norm of any dimension. The theoretical results are validated further bythe numerical tests in both 2D and 3D situations.