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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 76 (1997), S. 209-230 
    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
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  • 2
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 36 (1993), S. 539-567 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper addresses the problem of assessing the quality of an a posteriori error estimate of a finite element solution. An error estimate based on local L2-projections is analysed in the case of translation-invariant meshes. It is shown that for general meshes this technique does not lead to an asymptotically exact estimator. The problem is analysed in detail in the one-dimensional setting. It is shown that an asymptotically exact estimator is not the optimal one when the solution is not sufficiently smooth. An optimal estimator for adaptively constructed meshes is given. Finally, a general mathematical framework for the quality assessment of estimators is introduced.
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
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