A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908284994191360 |
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| author | Pechersky, David |
| author_facet | Pechersky, David |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps Pechersky, David Probability Analysis of PDEs Complex Variables Primary: Probability (math.PR), Secondary: Complex Variables (math.CV) 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. |
| title | A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps |
| topic | Probability Analysis of PDEs Complex Variables Primary: Probability (math.PR), Secondary: Complex Variables (math.CV) |
| url | https://arxiv.org/abs/2503.20284 |