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

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.