  We investigate the Cauchy problem for elliptic operators with
$C^\infty$-coefficients at a regular set $\Omega \subset R^2$, which is a
classical example of an ill-posed problem. The Cauchy data are given at the
subset $\Gamma \subset \partial\Omega$ and our objective is to reconstruct the
trace of the $H^1(\Omega)$ solution of an elliptic equation at $\partial \Omega
/ \Gamma$. The method described here is a generalization of the algorithm
developed by Maz'ya et al. [Ma] for the Laplace operator, who proposed a method
based on solving successive well-posed mixed boundary value problems (BVP)
using the given Cauchy data as part of the boundary data. We give an
alternative convergence proof for the algorithm in the case we have a linear
elliptic operator with $C^\infty$-coefficients. We also present some numerical
experiments for a special non linear problem and the obtained results are very
promisive.
