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A parametrix method for elliptic surface PDEs

Tristan Goodwill and Michael O’Neil

Vol. 7 (2025), No. 1, 171–217
Abstract

Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. We present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate fundamental solution, 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.

Keywords
surface elliptic PDE, Laplace–Beltrami, parametrix, surface boundary value problems
Mathematical Subject Classification
Primary: 35C15, 35J47, 35R01, 45B05, 65R20
Milestones
Received: 23 January 2024
Revised: 2 December 2024
Accepted: 31 December 2024
Published: 15 March 2025
Authors
Tristan Goodwill
Department of Statistics
University of Chicago
Chicago, IL
United States
Michael O’Neil
Courant Institute, NYU
New York, NY
United States