Smoothed Projections over Weakly Lipschitz Domains

TL;DR

This paper extends finite element exterior calculus to weakly Lipschitz domains by constructing commuting projections with uniform bounds, enabling analysis on more general polyhedral domains.

Contribution

It introduces new mathematical tools and constructs commuting projections for finite element de Rham complexes on weakly Lipschitz domains, broadening the scope of the theory.

Findings

Constructed commuting projections with uniform bounds.

Extended finite element differential forms to weakly Lipschitz domains.

Used collar theorem and geometric measure theory in the analysis.

Abstract

We develop finite element exterior calculus over weakly Lipschitz domains. Specifically, we construct commuting projections from $L^p$ de~Rham complexes over weakly Lipschitz domains onto finite element de~Rham complexes. These projections satisfy uniform bounds for finite element spaces with bounded polynomial degree over shape-regular families of triangulations. Thus we extend the theory of finite element differential forms to polyhedral domains that are weakly Lipschitz but not strongly Lipschitz. As new mathematical tools, we use the collar theorem in the Lipschitz category, and we show that the degrees of freedom in finite element exterior calculus are flat chains in the sense of geometric measure theory.