Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra

TL;DR

This paper introduces new fully discrete polynomial de Rham sequences on polygons and polyhedra, combining compatible discretizations and polyhedral methods, with proven exactness and practical implementation features.

Contribution

It presents novel polynomial de Rham sequences of arbitrary degree on polygons and polyhedra, with proven exactness and compatibility with finite element spaces.

Findings

Sequences are directly implementable in computer code.

Exactness of the constructed sequences is rigorously proven.

Compatibility with existing finite element spaces is established.

Abstract

In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving exactness, we show that the usual three-dimensional sequence of trimmed Finite Element spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. A discussion on reconstructions of potentials and discrete $L^2$-products completes the exposition.