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Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics

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Abstract

Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve Poisson’s equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N 2logN) operations on an N×N uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs.

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Author notes

  1. Graeme Fairweather

    Present address: Mathematical Reviews, American Mathematical Society, 416 Fourth Street, Ann Arbor, MI, 48103, USA

Authors and Affiliations

  1. Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO, 80401-1887, USA

    Bernard Bialecki & Graeme Fairweather

  2. Army G1 Military Personnel Structure and Plans Division, DAPE-PRP, Pentagon, Washington, DC, 20310-0300, USA

    David B. Knudson

  3. 1013 S. 47th St., Philadelphia, PA, 19081, USA

    D. Abram Lipman

  4. Avanade Inc., 2211 Elliott Avenue, Seattle, WA, 98121, USA

    Que N. Nguyen

  5. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Weiwei Sun

  6. 8179 Edge Rock Way D306, Laurel, MD, 20724, USA

    Gadalia M. Weinberg

Authors
  1. Bernard Bialecki
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  2. Graeme Fairweather
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  3. David B. Knudson
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  4. D. Abram Lipman
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  5. Que N. Nguyen
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  6. Weiwei Sun
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  7. Gadalia M. Weinberg
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Corresponding author

Correspondence to Graeme Fairweather.

Additional information

D. Abram Lipman and Gadalia M. Weinberg were supported in part by National Science Foundation grant DMS-9912293.

Que N. Nguyen was supported in part by National Science Foundation grant DGE-0231611.

Weiwei Sun was supported in part by a grant from City University of Hong Kong (Project No. CityU 7002110).

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Bialecki, B., Fairweather, G., Knudson, D.B. et al. Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics. Numer Algor 52, 1–23 (2009). https://doi.org/10.1007/s11075-008-9255-y

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  • DOI: https://doi.org/10.1007/s11075-008-9255-y

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