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A parametrix method for elliptic surface PDEs

Tristan Goodwill and Michael O’Neil
Vol. 7 (2025), No. 1, 171–217
DOI: 10.2140/paa.2025.7.171
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.

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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
© 2025 MSP (Mathematical Sciences Publishers).