  Having in mind the modelling of marble degradation under chemical pollutants,
e.g.~the sulfation process, we consider governing nonlinear diffusion equations
and their numerical approximation.The space domain of a computation is the
pristine marble object. In order to accurately discretize it while maintaining
the simplicity of finite difference discretizations, the domain is described
using a level-set technique. A uniform Cartesian grid is laid over a box
containing the domain, but the solution is defined and updated only in the grid
nodes that lie inside the domain, the level-set being employed to select them
and to impose accurately the boundary conditions. We use a Crank-Nicolson
scheme in time, while for the space variables the discretization is performed
by a standard Finite-Difference scheme for grid points inside the domain and by
a ghost-cell technique on the ghost points (by using boundary conditions). The
solution of the large nonlinear system is obtained by a Newton-Raphson
procedure and a tailored multigrid technique is developed for the inner linear
solvers. The numerical results, which are very satisfactory in terms of
reconstruction quality and of computational efficiency, are presented and
discussed at the end of the paper.
