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
