I have recently learned about Rodin and Sullivan's work that proved a conjecture of Thurston involving giving a construction for the map in the Riemann mapping theorem using circle packings and this prompted me to wonder about the possibility of a similar sort of story for harmonic functions and the Dirichlet problem.


Let $D$ be the unit disk in the plane. Then given any continuous real-valued function $f$ on the boundary of $D$, there exists a unique harmonic extension $F$ of $f$ to $D$.


For $G$ a connected graph there is a similar condition, given any real-valued function $g$ on some nonempty subset of the vertices vertices of $G$, $S \subset V(G)$ , there is a unique function $G : V(G) \to \mathbb{R}$ such that $G$ extends $g$ and is harmonic on all of the vertices not in $S$.


Namely, let $G\_\epsilon$ be the square grid in the plane, parallel to the coordinate axes with edge length $\epsilon$ and let $D\_\epsilon$ be the intersection of this infinite planar graph with $D$. Using $f$, we can assign values to the boundary vertices of $D\_\epsilon$ (say project radially to the boundary and take the value there) which we can then extend to give a harmonic function on $F\_\epsilon : V(D\_\epsilon) \to \mathbb{R}$.



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> Do these functions $F\_\epsilon$ "converge" to $F$ as $\epsilon \to 0$?
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The graphs $D\_\epsilon$ are embedded in the plane, so that gives one way of making sense of convergence. A similar story could maybe apply also in higher dimensions, I'd be happy to hear about that as well.


(Maybe this is easier for the honeycomb tiling?)



Harmonic functions as limits of harmonic functions on graphs?
Such approximations have a long history, starting with


[1] Courant, R., Friedrichs, K. and Lewy, H. (1928) Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 32–74


Good error estimates are in


[2] Laasonen, P. (1967) On the discretization error of the Dirichlet problem in a plane region with corners, Ann. Acad. Sci. Fenn. A I 408 2–16.


and the more delicate estimates near the boundary are at


[3] Kesten, Harry. "Relations between solutions to a discrete and continuous Dirichlet problem." In Random walks, Brownian motion, and interacting particle systems, pp. 309-321. Birkhäuser, Boston, MA, 1991.


One quick approach is to recall that the discrete Dirichlet problem is solved using random walk, the continuous one can be solved using Brownian motion, and Donsker's theorem then yields the convergence.


