  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.
