Construction of polynomial preserving cochain extensions by blending

TL;DR

This paper generalizes polynomial preserving boundary extensions from scalar functions to differential forms within the de Rham complex, ensuring they are cochain maps that commute with the exterior derivative.

Contribution

It introduces a blending technique to construct polynomial preserving cochain extensions for differential forms on simplices, extending classical scalar methods.

Findings

Extensions map boundary traces to interior forms

Extensions commute with exterior derivative

Generalizes scalar blending to differential forms

Abstract

A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an $n$ simplex to the interior is to use so-called rational blending functions. The purpose of this paper is to generalize the construction by blending to the de Rham complex. More precisely, we define polynomial preserving extensions which map traces of $k$ forms defined on the boundary of the simplex to $k$ forms defined in the interior. Furthermore, the extensions are cochain maps, i.e., they commute with the exterior derivative.