A stabilised finite element method for a fictitious domain problem allowing small inclusions

The purpose of this work is to approximate numerically an elliptic partial differential equation posed on domains with small perforations (or inclusions). The approach is based on the fictitious domain method, and since the method's interest lies in the case in which the geometrical features are not resolved by the mesh, we propose a stabilised finite element method. The stabilisation term is a simple, non-consistent penalisation, that can be linked to the Barbosa-Hughes approach. Stability and optimal convergence are proved, and numerical results confirm the theory.