Trimmed Serendipity Finite Element Differential Forms

TL;DR

This paper introduces a new family of finite element differential form spaces called trimmed serendipity spaces, applicable on cubical meshes in any dimension and degree, with proven unisolvent degrees of freedom and minimal compatibility.

Contribution

It defines and analyzes the properties of trimmed serendipity finite element differential form spaces, extending previous work on serendipity families and providing explicit degrees of freedom.

Findings

Spaces are defined on cubical meshes in any dimension.

Degrees of freedom are proven to be unisolvent.

Spaces form minimal compatible finite element systems.

Abstract

We introduce the family of trimmed serendipity finite element differential form spaces, defined on cubical meshes in any number of dimensions, for any polynomial degree, and for any form order. The relation between the trimmed serendipity family and the (non-trimmed) serendipity family developed by Arnold and Awanou [Math. Comp. 83(288) 2014] is analogous to the relation between the trimmed and (non-trimmed) polynomial finite element differential form families on simplicial meshes from finite element exterior calculus. We provide degrees of freedom in the general setting and prove that they are unisolvent for the trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a fixed polynomial order r provides an explicit example of a system described by Christiansen and Gillette [ESAIM:M2AN 50(3) 2016], namely, a minimal compatible finite element system on squares or cubes containing order r-1 polynomial differential forms.