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Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I

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Summary

This paper describes a fast and numerically stable method for solving the discrete Dirichlet problem for Poisson's equation in case of a rectangle (and mainly, a square). By using a special calculus for difference operators, the system of linear equations is reduced to a block-triangular system such that the diagonal blocks are heavily diagonally dominant. For a standard version of the algorithm, the number of operations and the computing time are proportional toh −2 (h=mesh width). The method is one oftotal reduction compared with the method ofblock-cyclic reduction (odd-even reduction) [2], which we describe as a method ofpartial reduction.—Due to the developed calculus, many generalizations are possible.—In a following part II of the paper, the algorithm and numerical results will be described in detail.

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Authors and Affiliations

  1. Mathematisches Institut der Universität Köln, Wevertal 86-90, D-5000, Köln 41, Bundesrepublik Deutschland

    Johann Schröder & Ulrich Trottenberg

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  1. Johann Schröder
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  2. Ulrich Trottenberg
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Schröder, J., Trottenberg, U. Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I. Numer. Math. 22, 37–68 (1974). https://doi.org/10.1007/BF01436620

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  • DOI: https://doi.org/10.1007/BF01436620

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