Restricted Element-Wise Projection for the Finite Element Method

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 preand post-processing like visualisation. We provide local and global L-error estimates along with numerical experiments verifying the theoretical results.