Publication Date:
2016-07-13
Description:
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 $L^\infty (L^p)$ for $p=1,\infty $ under the sole assumption of the initial datum being in $L^p$ . An improved bound for the error in $L^\infty (L^1)$ 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 behaviour as $t\to \infty $ . We finally present two examples on fractals: the Sierpinski gasket and the Sierpinski carpet, which illustrate on the effect of nonlocal diffusion for piece-wise constant initial datum.
Print ISSN:
0272-4979
Electronic ISSN:
1464-3642
Topics:
Mathematics
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