Mollification in strongly Lipschitz domains with application to continuous and discrete De Rham complex

TL;DR

This paper develops mollification operators in strongly Lipschitz domains that are stable, commute with differential operators, and are used to construct finite element projections with optimal approximation properties.

Contribution

It introduces mollification operators that do not require non-trivial extensions, commute with key differential operators, and facilitate finite element space projections in Lipschitz domains.

Findings

Constructed $L^p$ stable mollification operators in Lipschitz domains.

Characterized kernels of trace operators for vector fields.

Built projection operators with optimal approximation properties.

Abstract

We construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are $L^p$ stable for any real number $p\in[1,\infty]$, and commute with the differential operators $\nabla$, $\nabla{\times}$, and $\nabla{\cdot}$. We also construct mollification operators satisfying boundary conditions and use them to characterize the kernel of traces related to the tangential and normal trace of vector fields. We use the mollification operators to build projection operators onto general $H^1$-, $\mathbf{H}(\text{curl})$- and $\mathbf{H}(\text{div})$-conforming finite element spaces, with and without homogeneous boundary conditions. These operators commute with the differential operators $\nabla$, $\nabla{\times}$, and $\nabla{\cdot}$, are $L^p$-stable, and have optimal approximation properties on smooth functions.