Different approaches to the numerical solution of the 3D Poisson equation implemented in Python
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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.
KeywordsPoisson Equation Finite element method Python Factorization approach
Mathematics Subject Classification31-04 35J05 68N99
Financial support by the University of South Africa (UNISA) is acknowledged.
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