A new cubic nonconforming finite element on rectangles

TL;DR

This paper introduces a new cubic nonconforming finite element for rectangles that achieves cubic convergence, with a novel set of degrees of freedom and proven optimal error estimates, validated by numerical examples.

Contribution

It presents a new nonconforming finite element with specific degrees of freedom and proven optimal convergence for second-order elliptic problems on rectangles.

Findings

Achieves cubic convergence in energy norm

Provides optimal error estimates in energy and L2 norms

Numerical results confirm theoretical predictions

Abstract

A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the twelve values at the three Gauss points on each of the four edges. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eleven. The nonconforming element consists of $P_2\oplus \Span\{x^3y-xy^3\}$. We count the corresponding dimension for Dirichlet and Neumann boundary value problems of second-order elliptic problems. We also present the optimal error estimates in both broken energy and $L_2(\O)$ norms. Finally, numerical examples match our theoretical results very well.