A Tensor-Product Finite Element Cochain Complex with Arbitrary Continuity

TL;DR

This paper constructs tensor product finite element cochain complexes with arbitrary smoothness on Cartesian meshes, using a modified Hermite interpolation that commutes with the exterior derivative, enabling higher-dimensional extensions.

Contribution

It introduces a novel tensor product finite element cochain complex with arbitrary smoothness, based on a modified Hermite interpolation that ensures commuting properties.

Findings

Constructed a one-dimensional $C^m$-conforming finite element cochain complex.

Proved the complex commutes with the exterior derivative.

Extended the construction to higher dimensions via tensor products.

Abstract

We develop tensor product finite element cochain complexes of arbitrary smoothness on Cartesian meshes of arbitrary dimension. The first step is the construction of a one-dimensional $C^m$-conforming finite element cochain complex based on a modified Hermite interpolation operator, which is proved to commute with the exterior derivative by means of a general commutation lemma. Adhering to a strict tensor product construction we then derive finite element complexes in higher dimensions.