  In this note, we develop a new stabilization mechanism for cut finite element
methods that generalizes previous approaches of ghost penalty type in two ways:
(1) The quantity that is stabilized and (2) The choice of elements that are
connected in the stabilization. In particular, we can stabilize functionals of
the discrete function such as finite element degrees of freedom. We
subsequently show that the kernel of our ghost penalty operator defines a
finite element space based on discrete extensions in the spirit of those
introduced in Burman, E.; Hansbo, P. and Larson, M. G., CutFEM Based on
Extended Finite Element Spaces, arXiv2101.10052, 2021.
