Twelve Macro-Bubbles Enriched C$^1$-$P_k$ Quadrilateral Finite Elements

Source abstract

We show that there is no working $C^1$ polynomial finite elements on general quadrilateral meshes. It denies all such previous constructions by gathering vertex, edge, and internal polynomial basis functions. We construct, by adding the minimum number of 12 $P_k$ macro-bubbles, a family of $C^1$-$P_k$ finite elements for all $k\geq 6$. Numerical experiments by the $C^1$-$P_k$ elements, $6\leq k\leq 9,$ are performed.