When solving a (one-dimensional linear) boundary value problem by finite differences, how does the local and global error relate? I would have expected that if the local error was $O(h^k)$ in the step size $h$ then the global error would be $O(h^{k-1})$, as with integration of initial value problems (with the usual Lipschitz continuity condition). But when reading an example in Sauer's Numerical Analysis (section 7.2.1) where the second order central difference approximation $$ u''(x) = \frac{u(x - h) - 2 u(x) + u(x + h)}{h^2} + O(h^2) $$ is used, he seems to imply that the global error is $O(h^2)$ as well. Is that right, and if so why?
Possible follow-up: How does this generalize to partial differential equations?