A parametrix method for elliptic surface PDEs
Numerical Analysis
2025-03-19 v1 Numerical Analysis
Abstract
Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate Green's function, the surface PDEs are transformed into well-conditioned integral equations. We demonstrate high-order numerical examples of this method applied to problems on general surfaces using a variant of the fast multipole method based on smooth interpolation properties of the kernel. Lastly, we discuss extensions of the method to surfaces with boundaries.
Cite
@article{arxiv.2401.12501,
title = {A parametrix method for elliptic surface PDEs},
author = {Tristan Goodwill and Michael O'Neil},
journal= {arXiv preprint arXiv:2401.12501},
year = {2025}
}