ISSN:
0945-3245
Keywords:
Mathematics Subject Classification (1991): 65F10, 65G99, 65L10, 65L12, 65N22
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary. 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 $n \rightarrow \infty$ and $O(\tau^{\gamma$ )} as $\tau \rightarrow 0$ . 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s002110050260
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