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.

翻译：我们研究了紧致无边流形上散度型线性椭圆偏微分方程数值解的误差界。重点关注一类单调有限差分逼近方法，该方法通过强稳定性保证有界解的存在性。在包括Dirichlet问题在内的许多设定中，可以轻易证明所得解误差与格式的形式相容性误差成正比。然而我们观察到令人意外的现象：对于定义在紧致无边流形上的偏微分方程，这一结论未必成立。我们提出一类基于相容性误差为$O(h^\alpha)$的单调格式的特殊逼近方案。通过精巧构造障碍函数，我们证明了在维度$d$下解误差被$O(h^{\alpha/(d+1)})$所界定。同时给出一个具体算例，数值验证了该预测收敛阶。基于这些误差界，我们进一步设计了可证明收敛的解梯度逼近族。