This paper presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to establish the connection of vertex gradients across adjacent elements. Key features of the surface NZT finite element space include its $H^1$-relative conformity and weak $H({\rm div})$ conformity, allowing for stabilization without the use of artificial parameters. Under the assumption that the exact solution and the dual problem possess only $H^3$ regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a comprehensive analysis yielding optimal second-order convergence in the broken $H^1$ norm. Numerical experiments are provided to support the theoretical results.

翻译：本文提出了一种新颖的稳定化非协调有限元方法，用于求解表面双调和问题。该方法通过采用Piola变换建立相邻单元间顶点梯度的联系，将New-Zienkiewicz型（NZT）单元扩展至多面体（近似）表面。表面NZT有限元空间的关键特征包括其$H^1$相对协调性与弱$H({\rm div})$协调性，使得无需引入人工参数即可实现稳定化。在精确解及对偶问题仅具有$H^3$正则性的假设下，我们建立了能量范数下的最优误差估计，并首次通过全面分析，在分片$H^1$范数下获得了最优二阶收敛性。数值实验验证了理论结果。