A QUANTITATIVE DESCRIPTION OF MESH DEPENDENCE FOR THE DISCRETIZATION OF SINGULARLY PERTURBED NONCONVEX PROBLEMS

We investigate the limiting description for a finite-difference approximation of a singularly perturbed Allen-Cahn type energy functional. The key issue is to understand the interaction between two small length-scales: the interfacial thickness e and the mesh size of spatial discretization delta. Depending on their relative sizes, we obtain results in the framework of G-convergence for the (i) subcritical (epsilon >> delta), (ii) critical (epsilon similar to delta), and (iii) supercritical (epsilon << delta) cases. The first case leads to the same area functional as 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.