Trimmed Serendipity Finite Element Differential Forms
For researchers in finite element methods, this provides a new family of spaces that are minimal in a certain sense, extending prior work on serendipity elements.
The paper introduces trimmed serendipity finite element differential form spaces on cubical meshes in any dimension, for any polynomial degree and form order, providing unisolvent degrees of freedom. This yields a minimal compatible finite element system containing order r-1 polynomial differential forms.
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.