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

A finite difference method for the variational $p$-Laplacian