Rectangular $C^1$-$Q_k$ Bell finite elements in two and three dimensions

TL;DR

This paper introduces a new Bell type $C^1$-$Q_k$ finite element for rectangular meshes in 2D and 3D that maintains optimal convergence with reduced space complexity, suitable for higher polynomial degrees.

Contribution

The paper constructs a novel Bell type $C^1$-$Q_k$ finite element with reduced space complexity that retains optimal approximation order in 2D and 3D.

Findings

Retains optimal order of convergence

Significant reduction in space complexity

Numerical experiments confirm effectiveness

Abstract

Both the function and its normal derivative on the element boundary are $Q_k$ polynomials for the Bogner-Fox-Schmit $C^1$-$Q_k$ finite element functions. Mathematically, to keep the optimal order of approximation, their spaces are required to include $P_k$ and $P_{k-1}$ polynomials respectively. We construct a Bell type $C^1$-$Q_k$ finite element on rectangular meshes in 2D and 3D, which has its normal derivative as a $Q_{k-1}$ polynomial on each face, for $k\ge 4$. We show, with a big reduction of the space, the $C^1$-$Q_k$ Bell finite element retains the optimal order of convergence. Numerical experiments are performed, comparing the new elements with the original elements.