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

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.