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.

翻译：光滑嵌入三维空间中椭圆问题出现在薄膜力学、电磁学（调和矢量场）和计算几何中。本文提出了一种基于参数法的积分方程方法，适用于多种变系数曲面椭圆问题。通过使用近似格林函数，曲面偏微分方程被转化为良态积分方程。我们展示了该方法应用于一般曲面问题的高阶数值示例，采用了基于核函数光滑插值特性的快速多极子算法变体。最后，我们讨论了该方法向含边界曲面的推广。