Randomized Projection Operators onto Piecewise Polynomial Spaces

TL;DR

This paper presents new computable projection operators onto piecewise polynomial spaces that achieve optimal approximation properties and improve finite element discretizations for rough data.

Contribution

Introduction of sampling-based projection operators with optimal approximation properties for piecewise polynomial spaces and finite element methods.

Findings

Operators exhibit near-optimal $L^2$ and $H^{-1}$ approximation.

Enable finite element discretizations with optimal convergence rates.

Applicable as smoothers for incomplete or rough data.

Abstract

We introduce computable projection operators onto piecewise polynomial spaces, defined via sampling and discrete least-squares polynomial approximations. The resulting mappings exhibit (almost) optimal approximation properties in $L^2$ and $H^{-1}$. As smoothers for incomplete or rough data, they yield computable finite element discretizations with optimal convergence rates.