# A lowest-order composite finite element exact sequence on pyramids

## TL;DR

This paper introduces lowest-order composite pyramidal finite elements that form an exact sequence respecting the de Rham diagram, ensuring compatibility and optimal approximation properties with existing tetrahedral and hexahedral elements.

## Contribution

The paper constructs the first lowest-order composite pyramidal finite elements that form an exact sequence and are compatible with standard tetrahedral and hexahedral finite elements.

## Key findings

- Elements respect the de Rham diagram and form an exact sequence.
- They are fully compatible with standard Raviart-Thomas-Nédélec elements.
- Achieve optimal approximation order comparable to pure tetrahedral or hexahedral meshes.

## Abstract

Composite basis functions for pyramidal elements on the spaces $H^1(\Omega)$, $H(\mathrm{curl},\Omega)$, $H(\mathrm{div},\Omega)$ and $L^2(\Omega)$ are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart-Thomas-N\'ed\'elec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, ...) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements is successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size $h$ as one would obtain using purely hexahedral or purely tetrahedral partitions.