Exact planar electrical flow
Using nested dissections you can even solve a linear system (based on a planar graph) in $O(\sqrt{n^\omega})$. This was for example noted in a paper  I have with Günter Rote and Ares Ribó and in this paper by Alon and Yuster. 

The former paper also contains an approach how to compute the pairwise resistances between vertices on a a common face in $O(\sqrt{n^\omega})$. This paper by Kenyon might also contain helpful ideas.