A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps

David Pechersky

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arXiv · 2503.20284

A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps

David Pechersky

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Abstract

We extend recent work of Gurel-Gurevich--Jerison--Nachmias (2020) and Bou-Rabee--Gwynne (2024) by showing that as the mesh of our lattice tends to $0$, we have a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps with H\"older boundary data to its continuous counterpart. The key idea is that the convolution of a discrete harmonic function on an orthodiagonal map with a smooth mollifier has small Laplacian and so is ``almost harmonic." This also allows us to show that discrete harmonic functions on orthodiagonal maps are Lipschitz in the bulk on a mesoscopic scale.

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David Pechersky. 2025-03-26. A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps. https://arxiv.org/abs/2503.20284

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