Higher-order finite element de Rham complexes, partially localized flux reconstructions, and applications

TL;DR

This paper develops higher-order finite element de Rham complexes with localized flux reconstructions, advancing adaptive methods and error estimation in finite element exterior calculus.

Contribution

It introduces a systematic construction of higher-order complexes and a new partially localized flux reconstruction technique for improved error estimation.

Findings

Constructed higher-order finite element de Rham complexes.

Developed a partially localized flux reconstruction method.

Generalized the Braess-Schöberl error estimator for higher-order edge elements.

Abstract

We construct finite element de~Rham complexes of higher and possibly non-uniform polynomial order in finite element exterior calculus (FEEC). Starting from the finite element differential complex of lowest-order, known as the complex of Whitney forms, we incrementally construct the higher-order complexes by adjoining exact local complexes associated to simplices. We define a commuting canonical interpolant. On the one hand, this research provides a base for studying $hp$-adaptive methods in finite element exterior calculus. On the other hand, our construction of higher-order spaces enables a new tool in numerical analysis which we call "partially localized flux reconstruction". One major application of this concept is in the area of equilibrated a~posteriori error estimators: we generalize the Braess-Sch\"oberl error estimator to edge elements of higher and possibly non-uniform order.