Eecient Algorithms for Diiusion-generated Motion by Mean Curvature

The problem of simulating the motion of evolving surfaces with junctions according to some curvature-dependent speed arises in a number of applications. By alternately diiusing and sharpening characteristic functions for each region, a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums. However, the usual nite diierence discretization of these methods are often excessively slow when accurate solutions are sought, even in two dimensions. We propose a new, spectral discretization of these diiusion-generated methods which obtains greatly improved eeciency over the usual-nite diierence approach. These eeciency gains are obtained, in part, through the use of a quadrature-based reenement technique, by integrating Fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients. Indeed, numerical studies demonstrate that the new algorithm is often more than 1000 times faster than the usual nite diierence discretization. Our ndings are demonstrated on several examples.