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

David Pechersky

TL;DR

The paper proves a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps to its continuum counterpart as the mesh $\varepsilon\to0$, for Hölder boundary data. The core method combines mollification of discrete harmonic functions to produce almost-harmonic functions, Beurling-type estimates to control boundary-influenced behavior, and Green-function representations to compare discrete and continuous solutions. A principal contribution is the explicit, though not sharp, rate bound $|h^{\bullet}(v)-h(v)| \le C (\varepsilon/\mathrm{diam}(\Omega))^{\lambda(\alpha,\beta)}$ with an exponent $\lambda(\alpha,\beta)$ defined via a minimax expression, plus a robust mesoscopic Lipschitz regularity result for discrete harmonic functions on OD maps. The results extend prior work by Gurel-Gurevich–Jerison–Nachmias and Bou-Rabee–Gwynne, and provide a framework for quantitative convergence in discrete complex analysis on 2D lattice discretizations, with implications for critical lattice models. Overall, the paper advances the quantitative understanding of discretization convergence in planar domains and introduces tools that may improve rates via a bootstrap of mesoscopic regularity.

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 Hölder 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.

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

TL;DR

The paper proves a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps to its continuum counterpart as the mesh , for Hölder boundary data. The core method combines mollification of discrete harmonic functions to produce almost-harmonic functions, Beurling-type estimates to control boundary-influenced behavior, and Green-function representations to compare discrete and continuous solutions. A principal contribution is the explicit, though not sharp, rate bound with an exponent defined via a minimax expression, plus a robust mesoscopic Lipschitz regularity result for discrete harmonic functions on OD maps. The results extend prior work by Gurel-Gurevich–Jerison–Nachmias and Bou-Rabee–Gwynne, and provide a framework for quantitative convergence in discrete complex analysis on 2D lattice discretizations, with implications for critical lattice models. Overall, the paper advances the quantitative understanding of discretization convergence in planar domains and introduces tools that may improve rates via a bootstrap of mesoscopic regularity.

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 , we have a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps with Hölder 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.

Paper Structure

This paper contains 20 sections, 17 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps
  3. Lipschitz Regularity in the Bulk on a Mesoscopic Scale for Harmonic Functions on Orthodiagonal Maps
  4. Conventions and Notation
  5. A Brief Introduction to Orthodiagonal Maps
  6. The Main Result and an Outline of its Proof
  7. Proof Outline: $z$ Near the Boundary
  8. Proof Outline: $z$ Away from the Boundary
  9. The Convolution of a Discrete Harmonic Function with a Smooth Mollifier is Almost Harmonic
  10. Proof of Theorem \ref{['thm: polynomial rate of convergence for Dirichlet problem on OD maps']}
  11. Case 1: $z_{0}$ is close to $\partial{\Omega}$
  12. Case 2: $z_{0}$ is far away from $\partial{\Omega}$
  13. Choosing an Optimal $\delta$
  14. Improving our Rate of Convergence
  15. Lipschitz Regularity on a Mesoscopic Scale for Harmonic Functions on Orthodiagonal Maps
  16. ...and 5 more sections

Key Result

Proposition 2.1

If $G=(V,E,c)$ is a finite network, $U\subsetneq{V}$, and $f:V\rightarrow{\mathbb{R}}$ is harmonic on $U$. Define: Then:

Theorems & Definitions (28)

  • Proposition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 18 more