A Quantitative Description of Mesh Dependence for the Discretization of Singularly Perturbed Nonconvex Problems
Andrea Braides, Nung Kwan Yip · SIAM Journal on Numerical Analysis · 2012
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 $\varepsilon$ 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) subcritical ($\varepsilon\gg \delta$), (ii) critical ($\varepsilon \sim \delta$), and (iii) supercritical ($\varepsilon\ll\delta$) 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.