# Discrete solution to differential equation on a closed irregular surface mesh

I've want to efficiently (real time) model a simple differential equation on a surface mesh (low resolution). However while I can work out how to implement the 2D equivalent of this problem of a differential equation on the boundary of a closed set of line segments (i.e. a polygon) by treating it as an irregular finite difference problem with vertices as the sampling points and applying a periodic boundary condition. When I try to consider extending this to a 3d mesh, my reasoning would indicate that the structure of the mesh would alter the solution.

Consider a simple plane, we could apply a regular square grid to this surface, and solve it using normal finite difference, however if we were to convert this quad mesh to a triangle mesh, the vertices would be influences by some but not all of the diagonal vertices, and further more with the dimensions becoming non-separable, this doesn't seam to extending properly to higher order differential equations.

Since I haven't been able to find any thing useful online (quite possibly because I'm not really sure what to search for), I'm hoping someone here can help me resolve this problem. Note for other reasons (relating to other parts of my simulation), I need to treat my vertices as sample points, and a triangular mesh, further more the vertices will be connected to 3 or more edges with no actual upper limit.

I'm hoping some one can actually provide a example for a simple problem such as time dependent surface diffusion.

$$\frac{\partial \varphi}{\partial t} = D\,\left(\frac{\partial^2 \varphi}{\partial v^2} + \frac{\partial^2 \varphi}{\partial u^2}\right)$$

where u and v are the surface parameters, i.e. two orthogonal tangents to the surface.