# A Family of Crouzeix-Raviart Finite Elements in 3D

## TL;DR

This paper introduces a new family of non-conforming Crouzeix-Raviart finite elements in 3D, providing explicit basis functions, optimal estimates, and analysis of their properties on simplicial meshes.

## Contribution

It develops a novel 3D non-conforming finite element family with explicit basis functions and theoretical analysis, extending prior 2D concepts.

## Key findings

- Optimal a priori estimates established
- Explicit local basis functions derived
- Analysis of linear independence and spanning properties

## Abstract

In this paper we will develop a family of non-conforming "Crouzeix-Raviart" type finite elements in three dimensions. They consist of local polynomials of maximal degree $p\in\mathbb{N}$ on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space.