论文标题
关于有限差异方案的准确性降低没有边界的歧管
On the Reduction in Accuracy of Finite Difference Schemes on Manifolds without Boundary
论文作者
论文摘要
我们研究了在没有边界的紧凑型歧管上发散结构线性椭圆PDE的数值解的误差界限。我们的重点是一类单调有限差近似值，它们提供了强大的稳定性形式，可以保证存在有限解决方案。在包括Dirichlet问题在内的许多设置中，很容易证明所得的解决方案错误与该方案的形式一致性错误成正比。我们做出了令人惊讶的观察，即对于没有边界的紧凑型歧管上的PDE而言，这并不是正确的。我们提出了一类特定类别的近似方案，围绕一个基础单调方案构建，并具有一致性错误$ O（H^α）$。通过仔细构造屏障功能，我们证明解决方案错误在dimension $ d $中由$ o（h^{α/（d+1）}）$界定。我们还提供了一个具体的示例，其中该预测的收敛速率被数值观察到。使用这些误差边界，我们进一步设计了一个可证明的收敛近似族的家族。
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 $O(h^α)$. By carefully constructing barrier functions, we prove that the solution error 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.
