Kernel-Based Galerkin Methods on Compact Manifolds Without Boundary, with an Emphasis on SO(3)

Abstract

We develop a kernel-based Galerkin method for numerically solving elliptic partial differential equations on compact Riemannian manifolds without boundary that are \textit{mesh-free}, \textit{coordinate-free}, \textit{regularity-preserving}, and \textit{high-performing}. A fundamental challenge is to combine a theoretical solution with mesh-free \textit{quadrature} in such a way that approximation power is not lost.\\\indent We show that the approximation power of the computed (discretized) solution can be made to be on par with the approximation power of the theoretical solution, provided the \textit{oversampling exponent} is sufficiently large. We then show how to truncate a kernel on $ SO(3) $ given by a Hilbert-Schmidt series in such a way that the approximation power of both the truncated solution and the discretized truncated solution (again using quadrature) is on par with the approximation power of the theoretical solution, provided the \textit{truncation parameter} is sufficiently large.