In this work, we propose a projection operator based on restricted element-wise projections,
which can be applied to a large set of finite element spaces. In contrast to most other
projectors, such as the standard interpolation operator or the well-known Clément’s operator,
our projection operator does not utilise the node functionals defining the basis functions of
the finite element space. Moreover, our operator can be implemented as a modified version of
a standard assembly method, thus making it a ‘black-box’ algorithm, which does not require
more information about a finite element space than is already needed for the assembly of a
PDE discretisation. Important applications for our operator are its usage as a prolongation
and restriction operator for geometric multigrid methods as well as pre- and post-processing
like visualisation. We provide local and global L2-error estimates along with numerical experiments
verifying the theoretical results.