  We present a method of CutFEM type for the Poisson problem with either
Dirichlet or Neumann boundary conditions. The computational mesh is obtained
from a background (typically uniform Cartesian) mesh by retaining only the
elements intersecting the domain where the problem is posed. The resulting mesh
does not thus fit the boundary of the problem domain. Several finite element
methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The
originality of the present article consists in avoiding integration over the
elements cut by the boundary of the problem domain, while preserving the
optimal convergence rates, as confirmed by both the theoretical estimates and
the numerical results.
