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Fall 2020 Computing the Newton potential in the boundary integral equation for the Dirichlet problem of the Poisson equation
Wenchao Guan, Ying Jiang, Yuesheng Xu
J. Integral Equations Applications 32(3): 293-324 (Fall 2020). DOI: 10.1216/jie.2020.32.293


Evaluating the Newton potential is crucial for efficiently solving the boundary integral equation of the Dirichlet boundary value problem of the Poisson equation. In the context of the Fourier–Garlerkin method for solving the boundary integral equation, we propose a fast algorithm for evaluating Fourier coefficients of the Newton potential by using a sparse grid approximation. When the forcing function of the Poisson equation expressed in the polar coordinates has mth-order bounded mixed derivatives, the proposed algorithm achieves an accuracy of order 𝒪(n−m log3n), with requiring 𝒪(nlog2n) number of arithmetics for the computation, where n is the number of quadrature points used in one coordinate direction. With the help of this algorithm, the boundary integral equation derived from the Poisson equation can be efficiently solved by a fast fully discrete Fourier–Garlerkin method.


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Wenchao Guan. Ying Jiang. Yuesheng Xu. "Computing the Newton potential in the boundary integral equation for the Dirichlet problem of the Poisson equation." J. Integral Equations Applications 32 (3) 293 - 324, Fall 2020.


Received: 13 March 2019; Accepted: 22 June 2019; Published: Fall 2020
First available in Project Euclid: 17 September 2020

zbMATH: 07283059
MathSciNet: MR4150702
Digital Object Identifier: 10.1216/jie.2020.32.293

Primary: 31A30, 31C20

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


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Vol.32 • No. 3 • Fall 2020
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