Phase field method for mean curvature flow with boundary constraints (Q2838595)

Numerical approximations of the mean curvature flow are investigated occurring at the interfaces of width \(\epsilon\) evolving in a restricted area. The evolution of the mean curvature flow is closely related to the gradient flows, whereas also the energy has to be minimized. Numerical tests for the Allen-Cahn equation with penalized double well potential and Dirichlet boundary conditions suggest convergence rates \({\mathcal O} (\epsilon^2 (\ln (\epsilon))^2)\) which improve the observed convergence rates \({\mathcal O} (\epsilon (\ln (\epsilon))\) existing in the literature.