Thomson's principle for electrical networks states that if $G$ is a network (a weighted graph), $a$, $b$ are vertices of $G$, then the effective resistance between vertices $a$ and $b$ in $G$ is given by: 


$\mathcal{R}\_{eff}(a\leftrightarrow{b})=\inf\{\mathcal{E}(\theta): \theta \text{ is a flow from $a$ to $b$}, ||\theta||=1\}$


where 


$||\theta||=\sum\limits\_{y\sim{a}}\theta(a,y)$ 


is the strength of flow and $\mathcal{E}(\theta)$ is the energy of a flow:


$\mathcal{E}(\theta)=\sum\limits\_{\substack{\overrightarrow{e}\in{\overrightarrow{E}} \\ \overrightarrow{e}=(x,y)}}\theta(x,y)^{2}$


It's known that the energy- minimizing flow of unit strength is just the discrete gradient of the solution to the Dirichlet problem on $G$ with boundary conditions $1$ on $b$ and $0$ on $a$, however in my situation I can't solve this problem analytically and therefore compute the effective resistance explicitly. 


I was hoping however to construct a flow to get a good upper bound on my effective resistance. Can somebody link me to some papers where something like this is done? 


ps: The only technique I know of for constructing good flows is the method of random paths (see section 2.6 of: [Planar Maps, Random Walks and Circle Packings](https://arxiv.org/pdf/1812.11224.pdf)) however I have never seen this used to prove anything non- trivial.



Finding good flows to upper bound effective resistance
Well, if you graph is an approximation to something continuous, then you can cook up the flow from the corresponding solution in the continuum.


For example, Lyons' paper where he proves his criterion of transience contains such a construction for $\mathbb{Z}^3$. He considers the unit cubes with centers at vertices of $\mathbb{Z}^3$, so that each edge is intersecting one face of the cube. The flow through an edge is then defined to be the flux of $\nabla|z|^{-1}$ through the corresponding face. Because of harmonicity of $|z|^{-1}$, this is indeed a flow (from the origin to infinity), and since $(\nabla|z|^{-1})^2$ decays as $|z|^{-4}$, this flow has finite energy. Hence, by Lyons' criterion, the random walk on $\mathbb{Z}^3$ is transient.


