# An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

## TL;DR

This paper introduces a new nonconforming finite element scheme using piecewise cubic polynomials for the planar biharmonic equation, achieving optimal convergence on general triangulations with a novel basis construction.

## Contribution

It develops a piecewise cubic nonconforming finite element scheme with proven optimal convergence and a novel basis construction for the biharmonic equation on general triangulations.

## Key findings

- Achieves $ ext{O}(h^2)$ convergence rate in energy norm.
- Establishes an equivalence between discrete Hessian and Laplacian forms.
- Constructs a locally supported basis without Ciarlet's triple.

## Abstract

This paper presents a nonconforming finite element scheme for the planar biharmonic equation which applis piecewise cubic polynomials ($P_3$) and possesses $\mathcal{O}(h^2)$ convergence rate in energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the $H^4$ functions with $\mathcal{O}(h^2)$ accuracy in broken $H^2$ norm. Besides, an equivalence $(\nabla_h^2\ \cdot,\nabla_h^2\ \cdot)=(\Delta_h\ \cdot,\Delta_h\ \cdot)$, which is usually not true for nonconforming finite element spaces, is proved on the newly designed spaces. The finite element space does not correspond to a finite element defined with Ciarlet's triple; however, a set of locally supported basis functions of the finite element space is still figured out. The notion of the finite element Stokes complex plays an important role in the analysis and also the construction of the basis functions.