A Convergent Scheme for Motion by Mean Curvature
- A convergent monotone difference scheme for motion of level sets by mean curvature, Numerische Mathematik, Volume 99 (2004) Number 2 pages 365-379.
This paper produced the first convergent finite difference scheme for motion of level sets by mean curvature. Motion by mean curvature is a famous problem with a wide body of work in geometric analysis, followed by viscosity solutions. While there are well-known schemes for moving a single level set by mean curvature, this scheme correctly approximates the PDE, which moves every level set by mean curvature.
The scheme involved the median of the neighboring grid points, which is an usual formula to appear in a difference scheme. In addition, while the scheme is less accurate, it is an order of magnitude faster than the obvious (non-provably convergent) finite difference scheme, which is to simply plug in finite differences to the terms in the equation.
Related work
For background on building convergence finite difference schemes for viscosity solutions, refer to