AbstractWe construct mollification operators in strongly
Lipschitz domains that do not invoke non-trivial extensions, are
Lp stable for any real number ${p\in [1,\infty ]}$, and commute with the differential operators ∇,
${\nabla {\times }}$, 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 H1-,
${{H}(\mathrm {curl})}$- and
${{H}(\mathrm {div})}$-conforming
finite element spaces, with and without
homogeneous boundary conditions. These operators commute with the
differential operators ∇, ${\nabla {\times }}$, and ${\nabla {\cdot }}$, are Lp-stable, and have optimal approximation
properties on smooth functions.