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

Document Type

Article

Publication Date

5-1-2024

Abstract

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.

Identifier

85195422264 (Scopus)

Publication Title

IMA Journal of Numerical Analysis

External Full Text Location

https://doi.org/10.1093/imanum/drad048

e-ISSN

14643642

ISSN

02724979

First Page

1751

Last Page

1784

Issue

3

Volume

44

Grant

1751996

Fund Ref

National Science Foundation

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