An Efficient Direct Method to Solve the Three Dimensional Poisson’s Equation
Alemayehu Shiferaw, Ramesh Chand Mittal
DOI: 10.4236/ajcm.2011.14035   PDF   HTML     8,860 Downloads   17,079 Views   Citations


In this work, the three dimensional Poisson’s equation in Cartesian coordinates with the Dirichlet’s boundary conditions in a cube is solved directly, by extending the method of Hockney. The Poisson equation is approximated by 19-points and 27-points fourth order finite difference approximation schemes and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The efficiency of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results. It is shown that 19-point formula produces comparable results with 27-point formula, though computational efforts are more in 27-point formula.

Share and Cite:

A. Shiferaw and R. Chand Mittal, "An Efficient Direct Method to Solve the Three Dimensional Poisson’s Equation," American Journal of Computational Mathematics, Vol. 1 No. 4, 2011, pp. 285-293. doi: 10.4236/ajcm.2011.14035.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] L. Collatz, “The Numerical Treatment of Differential Equa- tions,” Springer Verlag, Berlin, 1960.
[2] M. K. Jain, “Numerical Solution of Differential Equa- tions,” New Age International ltd, New Delhi, 1984.
[3] R. Haberman, “Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Prob- lems,” Prentice-Hall Inc., Saddle River, 1987.
[4] T. Myint-U and L. Debnath, “Linear Partial Differential Equations for Scientists and Engineers,” Birkhauser, Bos- ton, 2007
[5] G. D. Smith, “Numerical Solutions of Partial Differential Equations: Finite Difference Methods,” Oxford Univer- sity Press, New York, 1985.
[6] G. H. Golub and C. F. van Loan, “Matrix Computations,” Johns Hopkins University Press, Baltimore, 1989.
[7] J. Stoer and R. Bulirsch, “Introduction to Numerical Ana- lysis,” Springer-Verlag, New York, 2002.
[8] R. W. Hockney, “A Fast Direct Solution of Poisson Equa- tion Using Fourier Analysis,” Journal of ACM, Vol. 12, No. 1, 1965, pp. 95-113. doi:10.1145/321250.321259
[9] W. W. Lin, “Lecture Notes of Matrix Computations,” Na- tional Tsing Hua University, Hsinchu, 2008.
[10] B. L. Buzbee, G. H. Golub and C. W. Nielson, “On Di- rect Methods for Solving Poisson’s Equations,” SIAM Journal on Numerical Analysis, Vol. 7, No. 4, 1970, pp. 627-656. doi:10.1137/0707049
[11] A. Averbuch, M. Israeli and L. Vozovoi, “A Fast Pois- son’s Solver of Arbitrary Order Accuracy in Rectangular regions,” SIAM Journal on Scientific Computing, Vol. 19, No. 3, 1998, pp. 933-952. doi:10.1137/S1064827595288589
[12] A. McKenney, L. Greengard and A. Mayo, “A Fast Pois- son Solver for Complex Geometries,” Journal of Compu- tation Physics, Vol. 118, No. 2, 1996, pp. 348-355. doi:10.1006/jcph.1995.1104
[13] G. Skolermo, “A Fourier Method for Numerical Solution of Poisson’s Equation,” Mathematics of Computation, Vol. 29, No. 131, 1975, pp. 697-711.
[14] L. Greengard and J. Y. Lee, “A Direct Adaptive Poisson Solver of Arbitrary Order Accuracy,” Journal of Compu- tation Physics, Vol. 125, No. 2, 1996, pp. 415-424. doi:10.1006/jcph.1996.0103
[15] W. F. Spotz and G. F. Carey, “A High-Order Compact Formulation for the 3D Poisson Equation,” Numerical Methods for Partial Differential Equations, Vol. 12, No. 2, 1996, pp. 235-243. doi:10.1002/(SICI)1098-2426(199603)12:2<235::AID-NUM6>3.0.CO;2-R
[16] E. Braverman, M. Israeli, A. Averbuch and L. Vozovoi, “A Fast 3D Poisson Solver of Arbitrary Order Accuracy,” Journal of Computation Physics, Vol. 144, No. 1, 1998, pp. 109-136. doi:10.1006/jcph.1998.6001
[17] G. Sutmann and B. Steffen, “High Order Compact Solvers for the Three-Dimensional Poisson Equation,” Journal of Computation and Applied Mathematics, Vol. 187, No. 2, 2006, pp. 142-170. doi:10.1016/
[18] J. Zhang, “Fast and High Accuracy Multigrid Solution of the Three Dimensional Poisson Equation,” Journal of Computation Physics, Vol. 143, No. 2, 1998, pp. 449- 461. doi:10.1006/jcph.1998.5982
[19] M. A. Malcolm and J. Palmer, “A Fast Method for Solv- ing a Class of Tri-Diagonal Linear Systems,” Communi- cations of the ACM, Vol. 17, No. 1, 1974, pp. 14-17. doi:10.1145/360767.360777

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.