Résumé en anglais

We propose in this paper a new algorithm for computing the evolution by mean curvature of a hypersurface. Our algorithm is a variant of the variational approach of Almgren, Taylor and Wang~\cite{ATW}. We show that it approximates, as the time--step goes to zero, the generalized motion(in the sense of barriers or viscosity solutions). The results still hold for the Anisotropic Mean Curvature Motion, as long as the anisotropy is smooth.