Nonlocal diffusions on fractals. Qualitative properties and numerical approximations.

Abstract

We propose a numerical method to approximate the solution of a nonlocal diffusion problem on a general setting of metric measure spaces. These spaces include, but are not limited to, fractals, manifolds and Euclidean domains. We obtain error estimates in L ∞(L p ) for p = 1,∞ under the sole assumption of the initial datum being in L p . An improved bound for the error in L ∞(L 1 ) is obtained when the initial datum is in L 2 . We also derive some qualitative properties of the solutions like stability, comparison principles and study the asymptotic behavior as t → ∞. We finally present two examples on fractals: the Sierpinski gasket and the Sierpinski carpet, which illustrate on the effect of nonlocal diffusion for piecewise constant initial datum.