On the reduction in accuracy of finite difference schemes on manifolds without boundary

Brittany Froese Hamfeldt, Axel G.R. Turnquist

Research output: Contribution to journalArticlepeer-review


We investigate error bounds for numerical solutions of divergence structure linear elliptic partial differential equations (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. By carefully constructing barrier functions, we prove that the solution error is bounded by in dimension. 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.

Original languageEnglish (US)
Pages (from-to)1751-1784
Number of pages34
JournalIMA Journal of Numerical Analysis
Issue number3
StatePublished - May 1 2024

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Computational Mathematics
  • Applied Mathematics


  • compact manifolds
  • elliptic partial differential equations
  • error bounds
  • finite difference methods


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