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

Paper

Abstract

serendipity finite element is a sub-element of

BFS finite element such that the element remains

-continuous and includes all

polynomials. In other words, it is a minimum of

bubbles enriched

finite element. We enrich the

and

spaces by

and

-bubble functions, respectively. For all

, we enrich the

spaces exactly by

bubble functions. We show the uni-solvence and quasi-optimality of the newly defined

serendipity elements. Numerical experiments by the

serendipity elements,

, are performed.