  We propose a new monotone finite difference discretization for the
variational $p$-Laplace operator, \[ \Delta_p u=\text{div}(|\nabla
u|^{p-2}\nabla u), \] and present a convergent numerical scheme for related
Dirichlet problems. The resulting nonlinear system is solved using two
different methods: one based on Newton-Raphson and one explicit method.
Finally, we exhibit some numerical simulations supporting our theoretical
results.
  To the best of our knowledge, this is the first monotone finite difference
discretization of the variational $p$-Laplacian and also the first time that
nonhomogeneous problems for this operator can be treated numerically with a
finite difference scheme.
