, Volume 95, Supplement 1, pp 49–60 | Cite as

Different approaches to the numerical solution of the 3D Poisson equation implemented in Python

  • Moritz BraunEmail author


The numerical solution of the three-dimensional Poisson equation with Dirichlet boundary conditions, which is of importance for a wide field of applications in Computational Physics and Theoretical Chemistry is considered using the method of finite elements for a model problem. The direct, the iterative and the factorized direct methods for solving the corresponding linear system of equations are discussed and implemented in the scripting language Python making use of the numpy and pysparse extensions. The relative performance of the different approaches is compared and it is shown, that the factorized direct method is vastly superior for larger problem sizes. A formalism for implementing the Dirichlet boundary conditions in the factorization approach is derived and presented in some detail, since it is to the best of our knowledge new.


Poisson Equation Finite element method Python  Factorization approach 

Mathematics Subject Classification

31-04 35J05 68N99 



Financial support by the University of South Africa (UNISA) is acknowledged.


  1. 1. Accessed 16 Jan 2013
  2. 2. Accessed 16 Jan 2013
  3. 3. Accessed 16 Jan 2013
  4. 4.
    Parr R (1989) Density functional theory of atoms and molecules. Oxford University Press, OxfordGoogle Scholar
  5. 5.
    Froese-Fischer C (1977) The Hartree-Fock method for atoms: a numerical approach. Wiley, New YorkGoogle Scholar
  6. 6.
    Fogolari F, Brigo A, Molinari H (2002) The Poisson–Boltzmann equation for biomolecular electrostatics: a tool for structural biology. J. Mol. Recognit. 15:377–392Google Scholar
  7. 7.
    Schellingerhout N.W.: Factorizability in the numerical few-body problem Ph.D. thesis, University of Groningen, The Netherlands, (1995)Google Scholar
  8. 8.
    Berger RJF, Sundholm D (2005) A non-iterative numerical solver of Poisson and Helmholtz equations using high-order finite-element functions. Adv. Quantum Chem. 50:235Google Scholar
  9. 9.
    Solin P (2006) Partial differential equations and the finite element method. Wiley InterscienceGoogle Scholar
  10. 10.

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of South Africa PretoriaSouth Africa

Personalised recommendations