A stabilized nonconforming finite element method for the surface biharmonic problem

TL;DR

This paper introduces a new stabilized nonconforming finite element method for the surface biharmonic problem, extending NZT elements to polyhedral surfaces with optimal convergence properties, supported by theoretical analysis and numerical validation.

Contribution

It develops a stabilized nonconforming finite element method for surface biharmonic problems, extending NZT elements with optimal error estimates without artificial parameters.

Findings

Optimal error estimates in energy norm.

First comprehensive analysis showing second-order convergence in broken H^1 norm.

Numerical experiments confirm theoretical results.

Abstract

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.