Equivalent extensions of partial differential equations on surfaces

This paper gives a general framework for the computation of several classes of partial differential equations (PDEs) on hypersurfaces. The approach is to work in a sufficiently thin narrow band, surrounding the surface, in the embedding Euclidean space and appropriately extend the differential operators so that the solutions on the narrow band and surface are equivalent. The solutions in the narrow band are equivalent to the solutions on the surface as in the former are constant along surface normal extensions of the latter. Consequently, it is possible to use existing (high order) numerical methods developed on grids in Euclidean space to solve PDEs on surfaces, with narrow bands whose widths are a small constant multiple of uniform Cartesian grid spacing. We apply the formulation to PDEs that originate from variational principles defined on surfaces as well as Hamilton-Jacobi-Bellman equations on surfaces that are derived from optimal control problems. This framework also provides a systematic way for solving PDE's on the unstructured point clouds that are sampled from the surface.