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 ε and the mesh size of spatial discretization δ. Depending on their relative sizes, we obtain results in the framework of Γ-convergence for the (i) subcritical (ε ≫ δ), (ii) critical (ε ~ δ), and (iii) supercritical (ε ≪ δ) 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. © 2012 Society for Industrial and Applied Mathematics.