A Numerical Method for Solving Elliptic Equations on Real Closed Algebraic Curves and Surfaces

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.