Continuum limit ofp-Laplacian evolution problems on graphs:Lqgraphons and sparse graphs
Imad El Bouchairi, Jalal M. Fadili, Abderrahim Elmoataz · ESAIM Mathematical Modelling and Numerical Analysis · 2023
In this paper we study continuum limits of the discretizedp-Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This goes far beyond known results by handling much more general class of kernels, possibly singular, and graph sequences whose limit are the so-calledLq-graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e., with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of thep-Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.