Quadrant Tridiagonal Partitioning and Preconditioned Conjugate Gradient Method for Solving Elliptic Problems

• G. Molnárka
• S. Szabó
Chapter
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NNFM, volume 29)

Abstract

Quadarant diagonal partitioning (QDP) method is an iterative method for solving linear system proposed by D. J. Evans et al. (see ,,) appropriate for paralell implementation. Here we show that in case of second order elliptic problem, the classical, color and multicolor ordering technique  are applicable for further paralellization of QDP iterative methods. Moreover we give a direct algorithm with arithmetical complexity O(n) for solving a special quadrant tridiagonal matrix and using these results we propose a quadrant tridiagonal preconditioned (QTP) conjugate gradient method for solving second order elliptic problems. The numerical experiments presented here shows that this method in several cases are more effective than other PCG algorithm.

References

1. 
Evans, D. J., Hadjidimos, A. and Noutsos, D., The parallel solution of banded linear equations by the new quadrant interlocking factorization (I.Q.F) method. International Journal of Computational Mathematics, 9. (1981).Google Scholar
2. 
Evans, D. J., Parallel numerical algorithm for linear systems. In Parallel Processing Systems (ed D. Evans) Cambridge University Press, (1982).Google Scholar
3. 
Evans, D. J., Parallel S.O.R. iterative methods. Parallel Computing, 1(1), (1984).Google Scholar
4. 
O’Leary, D., Ordering schemes for parallel processing of certain mesh problems. SIAM J. Sci. Stat. Comp. 5, pp. 620–632.Google Scholar
5. 
Hackbusch, W., On the multigrid method applied to difference equations. Computing 20. pp. 291–306, (1978).Google Scholar
6. 
Iain S. Duff, Gerard A. Meurant, The effect of ordering on preconditioned conjugate gradient. BIT 29. pp. 635–657, (1989).Google Scholar
7. 
Axelsson, O., A survey of preconditioned iterative methods for linear system of algebraic equations. BIT, pp. 166–187, (1985).Google Scholar
8. 
Axelsson, O., Vasilevski, P.S., A survey of multilevel preconditioned iterative methods. BIT 29, pp. 69–93, (1989).Google Scholar

Authors and Affiliations

• G. Molnárka
• 1
• S. Szabó
• 1
1. 1.Department of Numerical AnalysisEötvös UniversityBudapestHungary

Personalised recommendations

Citechapter 