  We develop a general framework for construction and analysis of discrete
extension operators with application to unfitted finite element approximation
of partial differential equations. In unfitted methods so called cut elements
intersected by the boundary occur and these elements must in general by
stabilized in some way. Discrete extension operators provides such a
stabilization by modification of the finite element space close to the
boundary. More precisely, the finite element space is extended from the stable
interior elements over the boundary in a stable way which also guarantees
optimal approximation properties. Our framework is applicable to all standard
nodal based finite elements of various order and regularity. We develop an
abstract theory for elliptic problems and associated parabolic time dependent
partial differential equations and derive a priori error estimates. We finally
apply this to some examples of partial differential equations of different
order including the interface problems, the biharmonic operator and the sixth
order triharmonic operator.
