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.

翻译：我们提出了分数阶$p$-拉普拉斯算子的新渐近展开式，并精确计算了误差。该逼近在$p\in(1,\infty)$和$s\in(0,1)$的整个范围内成立，且误差不会随着$s\to1^-$而退化。对于广泛的$p$值（在远离零梯度点处表现更优），误差呈超二次收敛，并在大多数情况下达到最优。本文的核心思想之一是：分数阶$p$-拉普拉斯算子积分表示中的奇异部分行为类似于具有权重修正的局部$p$-拉普拉斯算子。基于此，我们重新审视了经典$p$-拉普拉斯算子的一个先前渐近展开式（其误差阶此前未知）。在前述结果的基础上，我们提出了具有显式权重的分数阶$p$-拉普拉斯算子的单调有限差分逼近，并获得了误差估计。最后，我们引入了$\mathbb{R}^d$中相关抛物问题的显式有限差分格式，证明了该格式在粘性解意义下的稳定性、单调性和收敛性。一个有趣的特征是：稳定性条件随初始数据正则性的提高而改善。