Nodal bases for the serendipity family of finite elements

TL;DR

This paper introduces a method to construct nodal basis functions for the serendipity family of finite elements across various orders and dimensions, facilitating their computational implementation.

Contribution

It develops a general approach using multivariate lower set interpolation to create nodal bases for serendipity finite elements of any order and dimension.

Findings

Explicit construction of nodal basis functions

Representation as linear combinations of tensor-product polynomials

Applicable to any order and dimension

Abstract

Using the notion of multivariate lower set interpolation, we construct nodal basis functions for the serendipity family of finite elements, of any order and any dimension. For the purpose of computation, we also show how to express these functions as linear combinations of tensor-product polynomials.