A new fictitious domain approach inspired by the extended finite element method

The purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by Moës, Dolbow and Belytschko in [Moes et all]. An optimal method is obtained thanks to an additional stabilization technique. Some {\it a priori} estimates are established and numerical experiments illustrate different aspects of the method. The presentation is made on a simple Poisson problem with mixed Neumann and Dirichlet boundary conditions. The extension to other problems or boundary conditions is quite straightforward