The closest point method (CPM) is an embedding method for solving partial differential equations on surfaces. The closest point method uses standard numerical approaches such as finite differences, finite element or spectral methods in order to solve the embedding partial differential equation (PDE) which is equal to the original PDE on the surface. The solution is computed in a band surrounding the surface in order to be computationally efficient. In order to extend the data off the surface, the closest point method uses a closest point representation. This representation extends function values to be constant along directions normal to the surface.
Contents
Definitions
Closest Point function: Given a surface
Closest point extension: Let
Closest point method
Initialization consists of these steps [EW]:
After initialization, alternate between the following two steps:
Banding
The surface PDE is extended into
Example: Heat equation on a circle
Using initial profile
Applications
The closest point method can be applied to various PDEs on surfaces. Reaction–diffusion problems on point clouds [RD], eigenvalue problems [EV], and level set equations [LS] are a few examples.