The Numerical Evaluation of Particular Solutions for Poisson's Equation

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Abstract
Three numerical methods are described for the evaluation of particular solutions of Poisson's equation. The first method uses the Newton potential, evaluating it using Gaussian quadrature following a judicious change of integration variables. The second method uses a Fourier sine series to evaluate a particular solution, based on using a Fourier sine series expansion of the inhomogeneous term of the Poisson equation. Both of these methods assume that the inhomogeneous term can be extended smoothly to a suitable region larger than that of the original domain of the differential equation. The third method assume the inhomogeneous term is approximated by a polynomial, and then the Poisson equation with this new inhomogeneous term is solved exactly for a particular solution.

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