On the unisolvence for the quasi-polynomial spaces of differential forms

TL;DR

This paper establishes that the unisolvent sets for quasi-polynomial differential form spaces on simplices are identical to those for polynomial spaces, using a novel approach that avoids Stokes' Theorem, aiding the development of stable discretizations.

Contribution

It proves unisolvence for quasi-polynomial differential form spaces without relying on Stokes' Theorem, extending tools for stable discretizations in convection-diffusion problems.

Findings

Unisolvent sets for quasi-polynomial and polynomial differential form spaces are identical.

A novel proof approach avoids using Stokes' Theorem.

Results facilitate exponentially-fitted discretizations for convection-diffusion equations.

Abstract

We consider quasi-polynomial spaces of differential forms defined as weighted (with a positive weight) spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in any spatial dimension is the same as the set of such functionals used for the polynomial spaces. The analysis in the quasi-polynomial spaces, however, is not standard and requires a novel approach. We are able to prove our results without the use of Stokes' Theorem, which is the standard tool in showing the unisolvence of functionals in polynomial spaces of differential forms. These new results provide tools for studying exponentially-fitted discretizations stable for general convection-diffusion problems in Hilbert differential complexes.