A Fourier method for the numerical solution of Poisson's equation

Author:
Gunilla Sköllermo

Journal:
Math. Comp. **29** (1975), 697-711

MSC:
Primary 65N10

MathSciNet review:
0371096

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Abstract: A method for the solution of Poisson's equation in a rectangle, based on the relation between the Fourier coefficients for the solution and those for the right-hand side, is developed. The Fast Fourier Transform is used for the computation and its influence on the accuracy is studied. Error estimates are given and the method is shown to be second order accurate under certain general conditions on the smoothness of the solution. The accuracy is found to be limited by the lack of smoothness of the periodic extension of the inhomogeneous term. Higher order methods are then derived with the aid of special solutions. This reduces the problem to a case with sufficiently smooth data. A comparison of accuracy and efficiency is made between our Fourier method and the Buneman algorithm for the solution of the standard finite difference formulae.

**[1]**O. BUNEMAN,*A Compact Non-Iterative Poisson Solver*, Rep. 294, Stanford University, Institute for Plasma Research, Stanford, Calif., 1969.**[2]**B. L. Buzbee, G. H. Golub, and C. W. Nielson,*On direct methods for solving Poisson’s equations*, SIAM J. Numer. Anal.**7**(1970), 627–656. MR**0287717****[3]**J. W. COOLEY, P. A. W. LEWIS & P. D. WELCH, "The fast Fourier transform and its applications,"*IEEE Trans. Education*, v. E-12, 1969, pp. 27-34.**[4]**R. W. HOCKNEY, "The potential calculation and some applications,"*Methods in Computational Physics*, v. 9, 1970, pp. 135-211.**[5]**J. B. ROSSER,*Fourier Series in The Computer Age*, Mathematics Research Center, Technical Summary Report #1401, University of Wisconsin, Madison, Wis., Feb. 1974.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1975-0371096-4

Keywords:
Poisson's equation,
Fast Fourier Transform

Article copyright:
© Copyright 1975
American Mathematical Society