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We propose a new asymptotic expansion for the fractional $p$-Laplacian withprecise computations of the errors. Our approximation is shown to hold in thewhole range $p\in(1,\infty)$ and $s\in(0,1)$, with errors that do notdegenerate 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 ofthe main ideas here is the fact that the singular part of the integralrepresentation of the fractional $p$-Laplacian behaves like a local$p$-Laplacian with a weight correction. As a consequence of this, we alsorevisit a previous asymptotic expansion for the classical $p$-Laplacian, whoseerror orders were not known.  Based on the previous result, we propose monotone finite differenceapproximations of the fractional $p$-Laplacian with explicit weights and weobtain the error estimates. Finally, we introduce explicit finite differenceschemes for the associated parabolic problem in $\mathbb{R}^d$ and show that itis stable, monotone and convergent in the context of viscosity solutions. Aninteresting feature is the fact that the stability condition improves with theregularity of the initial data.

Higher-order asymptotic expansions and finite difference schemes for the fractional $p$-Laplacian