Reduced models for the Poisson problem in perforated domains

We develop a fictitious domain method to approximate a Dirichlet problem on a domain with small circular holes (simply called a perforated domain). To address the case of many small inclusions or exclusions, we propose a reduced model based on the projection of the homogeneous Dirichlet boundary constraint on a finite dimensional approximation space. We analyze the existence of the solution of this reduced problem and prove its convergence towards the limit problem without holes. We next obtain an estimate of the gap between the solution of the reduced model and the solution of the full initial model with small holes, the convergence rate depending on the size of the inclusion and on the number of modes of the finite dimensional space. The numerical discretization of the reduced problem is addressed by the finite element method, using a computational mesh that does not fit to the holes. The approximation properties of the finite element method are analyzed by a-priori estimates and confirmed by numerical experiments. elliptic differential equations, small inclusions, asymptotic analysis, approximated numerical method.