A Simple Technique for Solving Partial Differential Equations on Surfaces

Many applications require the solution of time dependent partial differential equations (PDEs) on surfaces or more general manifolds. Methods for treating such problems include surface parameterization, methods on triangulated surfaces and embedding techniques. In particular, embedding techniques using level set representations have received recent attention due to their simplicity. Level set based methods have several limitations, however. These include the inability to naturally treat open surfaces or objects of codimension two or higher. Level set methods also typically lead to a degradation in the order of accuracy when solved on a banded grid.

This talk describes an approach based on the closest point representation of the surface which eliminates these and other limitations. A noteworthy feature of the method is that it is remarkably simple, requiring only minimal changes to the corresponding three-dimensional codes to treat the evolution of partial differential equations on surfaces.