I was reading up on the Dirichlet problem, and was truly hoping if anyone here has the time to help make me understand this a bit better. In particular, the question relates to harmonic maps. My understanding is that a A (smooth) map $\varphi\colon M\to N$ between Riemannian manifolds $M$ and $N$ is called harmonic if it is a critical point of the Dirichlet energy functional. But now, lets imagine $M$ is a triangle mesh, which is a topological disk, how can I find its harmonic map? To be more specific, suppose $f\colon M \to R^2$ such that $$\begin{cases} \Delta f(v_i)=0&&\forall v_i\notin\partial M\\ f(v_i)=g(v_i)&&\forall v_i\in\partial M, \end{cases}$$ where $g\colon \partial M \to R^2$ is given.
I vaguely recall once learning about random walks, would that work here? How would that work?
Thank you all, I appreciate it.
You should revise the problem statement; as given in the question, it says that $\Delta f=0$ on the boundary, which is not right.
A harmonic map into $\mathbb R^2$ with Euclidean metric is just a pair of real-valued harmonic functions. To find such functions in a discrete setting, you can use the finite element method. Or, if you are looking for approximation defined on vertices of the mesh, you can use the discrete Laplacian: $\Delta f(v)$ is the average of $f$ over all neighbors of $v$, minus $f(v)$. Setting the discrete Laplacian to zero at all interior points, and equating $f$ to $g$ at the boundary points, you get a large system of linear equations, which can be easily solved with existing numerical algebra routines.
There is also the iterative approach: fill the vertices with some values at first, and then update them, replacing each with the average of its neighbors.
Random walk is not a practical way to find the solution.