$C^1$-$Q_k$ serendipity finite elements on rectangular meshes

TL;DR

This paper introduces new $C^1$-$Q_k$ serendipity finite elements on rectangular meshes, enriching polynomial spaces with bubble functions, and demonstrates their mathematical properties and numerical performance.

Contribution

It defines and analyzes new $C^1$-$Q_k$ serendipity finite elements, including their construction, unisolvency, and quasi-optimality, with numerical validation.

Findings

Elements are unisolvent and quasi-optimal.

Numerical experiments confirm effectiveness for $4 \,\leq\, k \leq 8$.

Enrichment with bubble functions improves finite element properties.

Abstract

A $C^1$-$Q_k$ serendipity finite element is a sub-element of $C^1$-$Q_k$ BFS finite element such that the element remains $C^1$-continuous and includes all $P_k$ polynomials. In other words, it is a minimum of $Q_k$ bubbles enriched $P_k$ finite element. We enrich the $P_4$ and $P_5$ spaces by $9$ $Q_4$ and $11$ $Q_5$-bubble functions, respectively. For all $k\ge 6$, we enrich the $P_k$ spaces exactly by $12$ $Q_k$ bubble functions. We show the uni-solvence and quasi-optimality of the newly defined $C^1$-$Q_k$ serendipity elements. Numerical experiments by the $C^1$-$Q_k$ serendipity elements, $4\le k\le 8$, are performed.