Nonlocal $p$-Laplacian evolution problems on graphs
Provides theoretical guarantees for numerical approximations of nonlocal p-Laplacian problems on graphs, relevant for researchers in graph-based PDEs and numerical analysis.
The paper derives error estimates and convergence rates for discretized nonlocal p-Laplacian evolution problems on graphs, proving that solutions of discrete models converge to the continuous solution as the number of vertices grows.
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.