  In this work, we analyse the links between ghost penalty stabilisation and
aggregation-based discrete extension operators for the numerical approximation
of elliptic partial differential equations on unfitted meshes. We explore the
behavior of ghost penalty methods in the limit as the penalty parameter goes to
infinity, which returns a strong version of these methods. We observe that
these methods suffer locking in that limit. On the contrary, aggregated finite
element spaces are locking-free because they can be expressed as an extension
operator from well-posed to ill-posed degrees of freedom. Next, we propose
novel ghost penalty methods that penalise the distance between the solution and
its aggregation-based discrete extension. These methods are locking-free and
converge to aggregated finite element methods in the infinite penalty parameter
limit. We include an exhaustive set of numerical experiments in which we
compare weak (ghost penalty) and strong (aggregated finite elements) schemes in
terms of error quantities, condition numbers and sensitivity with respect to
penalty coefficients on different geometries, intersection locations and mesh
topologies.
