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Approximate polynomial preconditionings applied to biharmonic equations

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Abstract

Applying a finite difference approximation to a biharmonic equation results in a very ill conditioned system of equations. This paper examines the conjugate gradient method used with polynomial preconditioning techniques for solving such linear systems. A new approach using an approximate polynomial preconditioner is described. The preconditioner is constructed from a series approximation based on the Laplacian finite difference matrix. A particularly attractive feature of this approach is that the Laplacian matrix consists of far fewer non-zero entries than the biharmonic finite difference matrix. Moreover, analytical estimates and computational results show that this preconditioner is more effective (in terms of the rate of convergence and the computational work required per iteration) than the polynomial preconditioner based on the original biharmonic matrix operator. The conjugate gradient algorithm and the preconditioning step can be efficiently implemented on a vector super-computer such as the CDC CYBER 205.

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Authors and Affiliations

  1. Department of Mathematics, University of Alberta, Canada

    Yau Shu Wong

  2. Institute for Computational Mechanics in Propulsion, NASA Lewis Research Center, USA

    Yau Shu Wong

  3. Department of Mathematics, University of Alberta, Canada

    Hong Jiang

Authors
  1. Yau Shu Wong
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  2. Hong Jiang
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Additional information

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada Grant U0375; and in part by NASA (funded under the Space Act Agreement C99066G) while the author was visiting ICOMP, NASA Lewis Research Center.

The work of this author was supported by an Izaak Walton Killam Memorial Scholarship.

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Wong, Y.S., Jiang, H. Approximate polynomial preconditionings applied to biharmonic equations. J Supercomput 3, 125–145 (1989). https://doi.org/10.1007/BF00129846

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