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

Abstract

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 http://www.python.org making use of the numpy http://www.numpy.org and pysparse http://pysparse.sourceforge.net 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.