MESH-FREE METHODS AND BOUNDARY CONDITIONS Serge DUMONT LAMFA, CNRS UMR 6140, UPJV, 33, Rue Saint Leu, 80039 Amiens CEDEX, France serge.dumont@u-picardie.fr Olivier GOUBET LAMFA, CNRS UMR 6140, UPJV, 33, Rue Saint Leu, 80039 Amiens CEDEX, France olivier.goubet@u-picardie.fr Tuong HA-DUONG LMAC, UTC, BP 20529, 60205 Compi` egne CEDEX, France tuong.ha-duong@utc.fr Pierre VILLON Laboratoire Roberval, UTC, BP 20529, 60205 Compi` egne, France pierre.villon@utc.fr We perform here some mesh free methods to inhomogeneous Laplace equations. We prove the efficiency of those methods compare with classical ones, for one or two D case for numerics, and for 1D for theorical results. Keywords : Laplace equations; meshless methods; wavelets. AMS Subject Classification: 65N12, 35J20, 65T60 1. Introduction Consider a bounded connected domain Ω whose boundary ∂ Ω is smooth. Consider a standard elliptic PDE problem on that domain. When looking for a finite element approximation for that problem, we need first to mesh the domain Ω. It is stan- dard to observe that the approximation results depend on how accurate the mesh approximates the boundary ( 9 ). In this article, we are concerned with non standard methods where the knowledge of ∂ Ω is not required. Our study takes part into the mesh-free methods (see 2 and the references therein). Loosely speaking, let us explain on a simple example what is a mesh-free method: let us pretend that we want to analyze the properties of some function whose support is included in Ω with finite elements or wavelets. Either we construct some ad hoc 1