This paper is devoted to a new algorithm to solve elastostatics problems on any shape domain. Our approach is based on a wavelet-Galerkin method. Due to the form of the wavelet bases, the general domain is included in a square fictitious domain, and the initial problem is extended over this square. The boundary conditions are imposed by an augmented Lagrangian method. Numerical examples are presented. Comparisons are done with analytic solutions and with finite element computations.