From my reading, u seems to be a suitable function that approximates heat flow on the surface of a manifold?
$u$ is the function that describes how your quantity behaves/evolves in a certain field. In the paper, the quantity is the temperature or heat flux, I guess. However, most of the time there is no analytical solution/formula for $u$. This is where methods like Finite Elements (FEM) come into play. By discretizing your field, you can piecewise approximate your function $u$.
In your case, you would use your mesh, which is already a discretization of your surface. Your elements are the triangles and you need to define how the nodal quantities are interpolated inside each triangle. --- Here, linear interpolation is probably the way to go. Otherwise, you need to remesh your geometry or introduce additional nodes for higher-order approximations.
Then you have to assign to each Node/Vertex an initial value $u_0$ as written in the answer of gilgamec. Afterwards, you build and solve your finite element system and get the nodal distribution of $u$ that actually solves your equation or system of equations. The finer your mesh gets, the better the solution. Higher-order interpolations will also help with accuracy.
So $u$ or its nodal values are what you are actually looking for as lightxbulb said in his comment. It's your unknown quantity.
If this doesn't help, you might want to read some literature about the finite element method. Can't tell how helpful the following links are, but a short glimpse looked promising. You will see, that they use $u$ all over the place. So I hope one of them will help you:
- A gentle introduction to the Finite Element Method
- PE281 Finite Element Method Course Notes
- Introduction to the finite element method
- finite element analysis by hand
I also had a link to a good online tutorial similar to the last link I provided that helped me a lot in understanding the fundamentals. If I find the link, I ll add it to my answer.
Found the link I was referring to. Unfortunately, it is in german: