Nonlocal $p$-Laplacian evolution problems on graphs

TL;DR

This paper analyzes the numerical approximation of nonlocal p-Laplacian evolution problems on graphs, establishing bounds, convergence, and error estimates for discrete models approaching the continuous solution, including the limit as p tends to infinity.

Contribution

It provides new bounds, convergence rates, and error estimates for discretized nonlocal p-Laplacian problems on graphs, including the p→∞ limit.

Findings

Bound on the distance between trajectories with different kernels and initial data

Convergence of discrete graph solutions to continuous problem as vertices increase

Uniform convergence results as p approaches infinity

Abstract

In this paper we study numerical approximations of the evolution problem for the nonlocal $p$-Laplacian with homogeneous Neumann boundary conditions. First, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels and initial data). We then provide a similar bound for the case when one of the trajectories is discrete-in-time and the other is continuous. In turn, these results allow us to establish error estimates of the discretized $p$-Laplacian problem on graphs. More precisely, for networks on convergent graph sequences (simple and weighted graphs), we prove convergence and provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows. We finally touch on the limit as $p \to \infty$ in these approximations and get uniform convergence results.