Higher-order asymptotic expansions and finite difference schemes for the fractional $p$-Laplacian
A talk in the Nonlocal Equations: Analysis and Numerics 2023 series by
Félix del Teso from Madrid
| Abstract: | The aim of this talk is to introduce the topic of asymptotic expansions and finite difference schemes for the fractional $p$-Laplacian $(-\Delta)^s_p \phi(x)= \textrm{P.V.}\int_{|z|>0} |\phi(x)-\phi(x+y)|^{p-1}(\phi(x)-\phi(x+y)) \frac{\textrm{d} y}{|y|^{d+sp}}$ in the range $s\in(0,1)$ and $p>1$. In this collaboration with M. Medina and P. Ochoa, we first present the nonlocal counterpart of the asymptotic expansions introduced in [1] [del Teso, Lindgren; NoDEA 2021]\ for the $p$-Laplacian, and show them to be a monotone approximation of super-quadratic order in most cases. We also introduce finite difference discretizations of $(-\Delta)^s_p$ and apply them to study numerically the associated Cauchy problem $\partial_t u +(-\Delta)^s_pu=f.$ We show that the explicit scheme is stable, monotone and convergent in the context of viscosity solutions. An interesting feature is the fact that the stability condition strongly depends on the regularity of the initial condition. [1] del Teso, Félix; Lindgren, Erik; A mean value formula for the variational p-Laplacian. $\textit{NoDEA Nonlinear Differential Equations Appl.}$ , 28 (2021), no. 3, Paper No. 27, 33 pp. Within the CRC this talk is associated to the project(s): A7 |