##### Abstract

The one-dimensional discrete Poisson equation on a uniform grid with $n$
points produces a linear system of equations with a symmetric
positive-definite coefficient matrix. Hence, the conjugate gradient method
can be used, and standard analysis gives an upper bound of $O(n)$ on the
number of iterations required for convergence. This paper introduces a
systematically defined set of solutions dependent on a parameter $\beta$,
and for several values of $\beta$, presents exact analytic expressions for
the number of steps $k(\beta,\tau,n)$ needed to achieve accuracy $\tau$.
The asymptotic behavior of these expressions has the form $O(n^{\alpha})$
as $n \to \infty$ and $O(\tau^{\gamma})$ as $\tau \to \infty$. In
particular, two choices of $\beta$ corresponding to nonsmooth solutions
give $\alpha=0$, i.e., iteration counts independent of $n$; this is in
contrast to the standard bounds. The standard asymptotic convergence
behavior, $\alpha=1$, is seen for a relatively smooth solution. Numerical
examples illustrate and supplement the analysis.
(Also cross-referenced as UMIACS-TR-95-86)