Rectangular $C^1$-$P_k$ finite elements with $Q_k$-bubble enrichment

TL;DR

This paper introduces new $C^1$-$P_k$ finite elements with $Q_k$-bubble enrichment on rectangular meshes, demonstrating their unisolvency, continuity, and convergence through numerical tests for various polynomial degrees.

Contribution

It develops a family of $C^1$-$P_k$ finite elements enriched with $Q_k$ bubbles for rectangular meshes, ensuring unisolvency and optimal convergence.

Findings

Finite elements are unisolvent and $C^1$ continuous.

Numerical tests confirm quasi-optimal convergence.

Enrichment improves approximation properties for $k \\ge 4$.

Abstract

We enrich the $P_k$ polynomial space by $5$ ($k=4$), or $7$ ($k=5$), or 8 (all $k\ge 6$) $Q_k$ bubble functions to obtain a family of $C^1$-$P_k$ ($k\ge 4$) finite elements on rectangular meshes. We show the uni-solvency, the $C^1$-continuity and the quasi-optimal convergence. Numerical tests on the new $C^1$-$P_k$, $k=4,5,6,7$ and $8$, elements are performed.