Discrete approximation of a free discontinuity problem

We approximate by discrete Г-convergence a functional proposed by Mumford-Shah for a variational approach to image segmentation. Such a functional is first relaxed with a sequence of nonconvex functionals, which in turn, are dis-cretized by piecewise linear finite elements. Under a suitable relation between the relaxation parameter ∊and the meshsize h, the convergence of the discrete functionals and the compactness of any sequence of discrete minimizers are proved. The proof relies on the techniques of Г-convergence and on the properties of the Lagrange interpolation and Clement operators.