Nonstandard finite element de Rham complexes on cubical meshes

TL;DR

This paper introduces two operations, DoF-transfer and serendipity, to construct nonstandard finite element complexes on cubical meshes, enabling convergent methods with enhanced regularity for complex PDEs.

Contribution

It develops a framework for creating nonstandard finite element complexes using DoF-transfer and serendipity operations, expanding the types of elements and methods available.

Findings

Elements provide convergent, non-conforming methods for PDEs.

The methods satisfy a discrete Korn inequality.

Potential applications include Stokes, biharmonic, and elasticity problems.

Abstract

We propose two general operations on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of "nonstandard" convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity, eliminates interior bubble functions and degrees of freedom locally on each element without affecting edge degrees of freedom. These operations can be used independently or in tandem to create nonstandard complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show that the resulting elements provide convergent, non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality. We discuss potential benefits of applying these elements to Stokes, biharmonic and elasticity problems.