On the Unisolvence for the Quasi-Polynomial Spaces of Differential Forms

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. Numerical experiments for scalar convection-diffusion problems in 2D demonstrate the superior performance of the exponentially-fitted method based on P₂-Lagrange elements.