A stabilized nonconforming finite element method for the surface biharmonic problem

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.