On the Reduction in Accuracy of Finite Difference Schemes on Manifolds without Boundary

TL;DR

This paper studies how finite difference schemes for elliptic PDEs on manifolds without boundary can have reduced accuracy, providing new error bounds and convergence analysis for these schemes.

Contribution

It introduces a class of monotone schemes on manifolds, establishes a novel error bound of O(h^{α/(d+1)}), and designs convergent gradient approximation methods.

Findings

Solution error is bounded by O(h^{α/(d+1)}) on manifolds without boundary.

Numerical experiments confirm the predicted convergence rate.

New gradient approximation schemes are proven to converge.

Abstract

We investigate error bounds for numerical solutions of divergence structure linear elliptic PDEs on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of stability that guarantees the existence of a bounded solution. In many settings including the Dirichlet problem, it is easy to show that the resulting solution error is proportional to the formal consistency error of the scheme. We make the surprising observation that this need not be true for PDEs posed on compact manifolds without boundary. We propose a particular class of approximation schemes built around an underlying monotone scheme with consistency error $O(h^\alpha)$. By carefully constructing barrier functions, we prove that the solution error is bounded by $O(h^{\alpha/(d+1)})$ in dimension $d$. We also provide a specific example where this predicted convergence rate is observed numerically. Using these error bounds, we further design a family of provably convergent approximations to the solution gradient.