Inserted: 20 jan 2011
Last Updated: 30 aug 2012
Journal: SIAM J. Numer Anal.
We investigate the limiting description for a finite-difference approximation of a singularly perturbed Allen-Cahn type energy functional. The key is to understand the interaction between two small length-scales: the interfacial thickness $\epsilon$ and the mesh size of spatial discretization $\delta$. Depending on their relative sizes, we obtain results in the framework of $\Gamma$-convergence for the (i) sub-critical ($\epsilon>> \delta$), (ii) critical ($\epsilon \sim \delta$), and (iii) super-critical ($\epsilon<<\delta$) cases. The first case leads to the same area functional just like the spatially continuous case while the third gives the same result as that coming from a ferromagnetic spin energy. The critical case can be regarded as an interpolation between the two.
Keywords: Discrete energies, finite-differences, non-convex problems, mesh dependence