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.