A least squares approach to Whitney forms

TL;DR

This paper introduces a least squares method for constructing Whitney forms with weights, addressing support selection challenges and potential Runge phenomena in high-order numerical analysis.

Contribution

It proposes a novel least squares approach for Whitney forms, improving support selection and addressing Runge phenomena in high-order finite element methods.

Findings

Consistent with nodal literature

Addresses support selection challenges

Makes progress on Runge phenomenon

Abstract

In this work we describe and test the construction of least squares Whitney forms based on weights. If, on the one hand, the relevance of such a family of differential forms is nowadays clear in numerical analysis, on the other hand the selection of performing sets of supports (hence of weights) for projecting onto high order Whitney forms turns often to be a rough task. As an account of this, it is worth mentioning that Runge-like phenomena have been observed but still not resolved completely. We hence move away from sharp results on unisolvence and consider a least squares approach, obtaining results that are consistent with the nodal literature and making some steps towards the resolution of the aforementioned Runge phenomenon for high order Whitney forms.