  We construct a cut finite element method for the membrane elasticity problem
on an embedded mesh using tangential differential calculus. Both free membranes
and membranes coupled to 3D elasticity are considered. The discretization comes
from a Galerkin method using the restriction of 3D basis funtions (linear or
trilinear) to the surface representing the membrane. In the case of coupling to
3D elasticity, we view the membrane as giving additional stiffness
contributions to the standard stiffness matrix resulting from the
discretization of the three-dimensional continuum.
