Limits and consistency of nonlocal and graph approximations to the Eikonal equation

Abstract In this paper, we study a nonlocal approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where kernel in problem is properly scaled. Based on theory viscosity solutions, prove existence and uniqueness solutions both local problems, as well regularity properties these time space. We then derive error bounds between solution to that one, continuous forward Euler discretization. turn studying continuum limits problems defined random weighted graphs $n$ vertices. particular, establish if scale parameter decreases at an appropriate rate grows then, almost surely, converges uniformly step vanishes number vertices large.