The authors study the geometric motion of a front \(\Sigma(t)\) in \(\mathbb{R}^n\) that propagates in its normal direction with speed given by the sum of its mean curvature and a forcing term. According to the Landau-Ginzburg theory of phase transitions, \(\Sigma(t)\) can be recovered as the limit as \(\varepsilon\to 0\) of the zero level set \(\Sigma_\varepsilon(t)\) of the solution of a suitable singularly perturbed reaction-diffusion equation. Several such equations have been used for this purpose.
Here, the authors choose one with a double obstacle potential and a space-time dependent relaxation parameter. They derive optimal order interface error estimates for smooth evolutions. These have a local character for small time, in that they depend on the local magnitude of the relaxation parameter. Finally, some numerical simulations are presented to illustrate how the variable relaxation parameter can be used to deal with large curvatures, and ultimately in resolving singularities.