Surface Multigrid via Global Parametric Domain Simplification

Abstract

We present ParaMG, a surface multigrid method that restores classical multigrid structure on curved surfaces by expressing all hierarchy levels in a single globally consistent planar domain. Existing surface multigrid methods rely on composed local parameterizations or 3D projections for cross‐level transfer, requiring repeated per‐level local approximations that can degrade convergence. Our key insight is that a global conformal parameterization with cone singularities provides a shared coordinate system with low, controllable distortion, where prolongation reduces to exact planar barycentric interpolation. We construct the hierarchy directly in this domain using a seam‐aware simplification strategy, yielding sparse prolongation matrices with three entries per row. ParaMG converges in fewer V‐cycles than prior surface multigrid methods and delivers substantial speedups in applications with changing linear systems, including geometric flows, thin‐shell simulation, and polycube deformation.