On the Reduction in Accuracy of Finite Difference Schemes on Manifolds without Boundary
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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. By carefully constructing barrier functions, we prove that the solution error achieved by a scheme with consistency error 𝒪(h^α) is bounded by 𝒪(h^α/(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.
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