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In this paper, we study a non-local approximation of the time-dependent(local) Eikonal equation with Dirichlet-type boundary conditions, where thekernel in the non-local problem is properly scaled. Based on the theory ofviscosity solutions, we prove existence and uniqueness of the viscositysolutions of both the local and non-local problems, as well as regularityproperties of these solutions in time and space. We then derive error boundsbetween the solution to the non-local problem and that of the local one, bothin continuous-time and Backward Euler time discretization. We then turn tostudying continuum limits of non-local problems defined on random weightedgraphs with $n$ vertices. In particular, we establish that if the kernel scaleparameter decreases at an appropriate rate as $n$ grows, then almost surely,the solution of the problem on graphs converges uniformly to the viscositysolution of the local problem as the time step vanishes and the number vertices$n$ grows large.

Limits and consistency of non-local and graph approximations to the Eikonal equation