  We propose a new asymptotic expansion for the fractional $p$-Laplacian with
precise computations of the errors. Our approximation is shown to hold in the
whole range $p\in(1,\infty)$ and $s\in(0,1)$, with errors that do not
degenerate as $s\to1^-$. These are super-quadratic for a wide range of $p$
(better far from the zero gradient points), and optimal in most cases. One of
the main ideas here is the fact that the singular part of the integral
representation of the fractional $p$-Laplacian behaves like a local
$p$-Laplacian with a weight correction. As a consequence of this, we also
revisit a previous asymptotic expansion for the classical $p$-Laplacian, whose
error orders were not known.
  Based on the previous result, we propose monotone finite difference
approximations of the fractional $p$-Laplacian with explicit weights and we
obtain the error estimates. Finally, we introduce explicit finite difference
schemes for the associated parabolic problem in $\mathbb{R}^d$ and show that it
is stable, monotone and convergent in the context of viscosity solutions. An
interesting feature is the fact that the stability condition improves with the
regularity of the initial data.

Analysis of PDEs;Numerical Analysis;Numerical Analysis