# Some iterative Poisson solvers applied to numerical solution of the model fourth-order elliptic problem

Aplikace matematiky (1985)

• Volume: 30, Issue: 3, page 176-186
• ISSN: 0862-7940

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## Abstract

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The numerical solution of the model fourth-order elliptic boundary value problem in two dimensions is presented. The iterative procedure in which the biharmonic operator is splitted into two Laplace operators is used. After formulating the finite-difference approximation of the procedure, a formula for the evaluation of the transformed iteration vectors is developed. The Jacobi semi-iterative, Richardson and A.D.I. iterative Poisson solvers are applied to compute one transformed iteration vector. By the efficient use of the decomposition property of the corresponding iteration matrices, the fast Fourier transform algorithm needs to be applied twice in the evaluation of one iteration vector. The asymptotic number of operations for the sequential computation is $5{n}^{2}lo{g}_{2}n$, where ${n}^{2}$ is the number of interior grid points in the unit square. The result of$7\phantom{\rule{4pt}{0ex}}lo{g}_{2}\phantom{\rule{4pt}{0ex}}n$ parallel steps for the parallel computation on an SIMD machine with ${n}^{2}$ processors is so far the best one.

## How to cite

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Vajteršic, Marián. "Some iterative Poisson solvers applied to numerical solution of the model fourth-order elliptic problem." Aplikace matematiky 30.3 (1985): 176-186. <http://eudml.org/doc/15396>.

@article{Vajteršic1985,
abstract = {The numerical solution of the model fourth-order elliptic boundary value problem in two dimensions is presented. The iterative procedure in which the biharmonic operator is splitted into two Laplace operators is used. After formulating the finite-difference approximation of the procedure, a formula for the evaluation of the transformed iteration vectors is developed. The Jacobi semi-iterative, Richardson and A.D.I. iterative Poisson solvers are applied to compute one transformed iteration vector. By the efficient use of the decomposition property of the corresponding iteration matrices, the fast Fourier transform algorithm needs to be applied twice in the evaluation of one iteration vector. The asymptotic number of operations for the sequential computation is $5n^2 log_2 n$, where $n^2$ is the number of interior grid points in the unit square. The result of$7 \ log_2 \ n$ parallel steps for the parallel computation on an SIMD machine with $n^2$ processors is so far the best one.},
author = {Vajteršic, Marián},
journal = {Aplikace matematiky},
keywords = {fourth-order; biharmonic operator; Laplace operators; Jacobi semi- iterative; Richardson; A.D.I.; fast Fourier transform; SIMD machine; fourth-order; biharmonic operator; Laplace operators; Jacobi semi- iterative; Richardson; A.D.I.; fast Fourier transform; SIMD machine},
language = {eng},
number = {3},
pages = {176-186},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some iterative Poisson solvers applied to numerical solution of the model fourth-order elliptic problem},
url = {http://eudml.org/doc/15396},
volume = {30},
year = {1985},
}

TY - JOUR
AU - Vajteršic, Marián
TI - Some iterative Poisson solvers applied to numerical solution of the model fourth-order elliptic problem
JO - Aplikace matematiky
PY - 1985
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 30
IS - 3
SP - 176
EP - 186
AB - The numerical solution of the model fourth-order elliptic boundary value problem in two dimensions is presented. The iterative procedure in which the biharmonic operator is splitted into two Laplace operators is used. After formulating the finite-difference approximation of the procedure, a formula for the evaluation of the transformed iteration vectors is developed. The Jacobi semi-iterative, Richardson and A.D.I. iterative Poisson solvers are applied to compute one transformed iteration vector. By the efficient use of the decomposition property of the corresponding iteration matrices, the fast Fourier transform algorithm needs to be applied twice in the evaluation of one iteration vector. The asymptotic number of operations for the sequential computation is $5n^2 log_2 n$, where $n^2$ is the number of interior grid points in the unit square. The result of$7 \ log_2 \ n$ parallel steps for the parallel computation on an SIMD machine with $n^2$ processors is so far the best one.
LA - eng
KW - fourth-order; biharmonic operator; Laplace operators; Jacobi semi- iterative; Richardson; A.D.I.; fast Fourier transform; SIMD machine; fourth-order; biharmonic operator; Laplace operators; Jacobi semi- iterative; Richardson; A.D.I.; fast Fourier transform; SIMD machine
UR - http://eudml.org/doc/15396
ER -

## References

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10. R. W. Hockney, The potential calculation and some applications, Methods in computational physics 9 (1970), 135-211. (1970)
11. A. H. Sameh S. C. Chen D. J. Kuck, Parallel Poisson and biharmonic solvers, Computing, Vol. 17(1976), 219-230. (1976) Zbl0342.65070MR0438737
12. M. Vajteršic, 10.1007/BF02252095, Computing, Vol. 23 (1979), 171-178. (1979) Zbl0406.65046MR0619928DOI10.1007/BF02252095
13. M. Vajteršic, A fast parallel solving the biharmonic boundary value problem on a rectangle, Proc. of 1st European Conference on Parallel and Distributed Processing, Toulouse 1979, 136-141. (1979) Zbl0406.65046
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15. D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York 1971. (1971) Zbl0231.65034MR0305568

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