Numerical Approximations for a Nonlocal Evolution Equation

In this paper we study numerical approximations of continuous solutions to the nonlocal $p$-Laplacian type diffusion equation, $u_t (t,x) = \int_{\Omega} J(x-y)|u(t,y) - u(t,x)|^{p-2}(u(t,y) - u(t,x)) \, dy$. First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties of the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition as $t$ goes to infinity. Next, we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties. In addition, we investigate the limit as $p$ goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile. Finally, we present some numerical experiments that illustrate our results.

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