  Motivated by many applications in complex domains with boundaries exposed to
large topological changes or deformations, fictitious domain methods regard the
actual domain of interest as being embedded in a fixed Cartesian background.
This is usually achieved via a geometric parameterization of its boundary via
level-set functions. In this note, the a priori analysis of unfitted numerical
schemes with cut elements is extended beyond the realm of linear problems. More
precisely, we consider the discretization of semilinear elliptic boundary value
problems of the form $- \Delta u +f_1(u)=f_2$ with polynomial nonlinearity via
the cut finite element method. Boundary conditions are enforced, using a
Nitsche-type approach. To ensure stability and error estimates that are
independent of the position of the boundary with respect to the mesh, the
formulations are augmented with additional boundary zone ghost penalty terms.
These terms act on the jumps of the normal gradients at faces associated with
cut elements. A-priori error estimates are derived, while numerical examples
illustrate the implementation of the method and validate the theoretical
findings.
