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 $O(h^{\alpha })$. 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.

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关于无边界流形上有限差分格式精度的降低

我们研究无边界紧流形上发散结构线性椭圆偏微分方程 (PDE) 数值解的误差界。我们的重点是一类单调有限差分近似，它提供了一种强大的稳定性形式，保证了有界解的存在。在包括狄利克雷问题在内的许多设置中，很容易证明所得到的解误差与方案的形式一致性误差成正比。我们做出了令人惊讶的观察，即对于无边界紧流形上提出的偏微分方程来说，这不一定是正确的。我们提出了一类特定的近似方案，围绕具有一致性误差 $O(h^{\alpha })$ 的底层单调方案构建。通过精心构造屏障函数，我们证明解误差在$d$维度上以$O(h^{\alpha /(d+1)})$为界。我们还提供了一个具体的例子，其中以数值方式观察了预测的收敛率。使用这些误差界限，我们进一步设计了一系列可证明收敛的解梯度近似值。