High-order finite elements on pyramids. II: unisolvency and exactness

TL;DR

This paper completes the definition of high-order conforming finite elements on pyramids by establishing unisolvency, compatibility, and polynomial inclusion, ensuring their integration into the de Rham complex.

Contribution

It introduces degrees of freedom for finite elements on pyramids, proving unisolvency, compatibility with other elements, and polynomial exactness, advancing finite element theory.

Findings

Elements are unisolvent and compatible with tetrahedral and hexahedral elements.

Elements satisfy a commuting diagram property.

Shape functions are tabulated for each element.

Abstract

We present degrees of freedom to accompany the approximation spaces already presented in a companion paper and thus complete the definition of families of high-order conforming finite elements on pyramids for the spaces of the de Rham complex. We prove that the elements are unisolvent; are compatible with conventional tetrahedral and hexahedral elements; satisfy a commuting diagram property and contain high-degree polynomials. We also tabulate shape functions for each element.