There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability between finite element method and computer aided design (CAD) software. However, these approaches have difficulty when the domain has singularities since the solution at the singularity may be multivalued. This paper develops a novel numerical approach to solve elliptic PDEs on real, closed, connected, orientable, and almost smooth algebraic curves and surfaces. Our method integrates numerical algebraic geometry, differential geometry, and a finite difference scheme which is demonstrated on several examples.

翻译：针对流形上偏微分方程的求解，已有多种数值方法，例如经典的隐式方法、有限差分法、有限元法以及等几何分析方法——后者旨在提升有限元法与计算机辅助设计软件之间的互操作性。然而，当求解域存在奇点时，这些方法面临困难，因为奇点处的解可能呈现多值性。本文提出一种新颖的数值方法，用于求解定义在实闭、连通、可定向且几乎光滑的代数曲线与曲面上的椭圆型偏微分方程。该方法融合了数值代数几何、微分几何及有限差分格式，并通过多个算例验证其有效性。