Integral Methods for Surface PDEs

Elliptic PDEs defined on surfaces occur frequently in computer graphics, and in many areas of physics, including electromagnetism, fluid dynamics, and molecular dynamics. Despite this broad range of applications, there are few efficient and accurate numerical methods for this class of PDEs. In this talk, I present a method for converting PDEs of this type into an integral equation over the surface. Unlike PDEs on surfaces, integral equations on surfaces have been thoroughly studied and existing tools may be used to construct fast and high-order numerical solvers.
In order to generate this integral equation form, I use a parametrix. Specifically, I observe that the Green’s function for a related PDE in the plane is a parametrix for the surface PDE in question. Once the regularity properties of this parametrix are discussed, I use it to derive the desired integral equation form and prove that this form is Fredholm second kind. Finally, I discuss a numerical method for solving the resulting integral equation.