The serendipity family of finite elements

TL;DR

This paper introduces a new, simple, dimension-independent definition of the serendipity finite element family, providing a clear characterization of shape functions and degrees of freedom, along with proofs of unisolvence and geometric decomposition.

Contribution

It offers a novel, unified definition of serendipity finite elements applicable in any dimension, with explicit shape functions and degrees of freedom.

Findings

Provides a dimension-independent definition of serendipity elements

Establishes unisolvence of the proposed finite element space

Offers a geometric decomposition of the space

Abstract

We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.