#### Document Type

Article

#### Publication Date

2020

#### DOI

10.1216/jie.2020.32.293

#### Publication Title

Journal of Integral Equations and Applications

#### Volume

32

#### Issue

3

#### Pages

293-324

#### Abstract

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 *m*th-order bounded mixed derivatives, the proposed algorithm achieves an accuracy of order πͺ(n^{-m} log^{3} *n*), with requiring πͺ(*n* log^{2} *n*) 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.

#### Original Publication Citation

Guan, W., Jiang, Y., & Xu, Y. (2020). Computing the Newton potential in the boundary integral equation for the Dirichlet problem of the Poisson equation. *Journal of Integral Equations and Applications, 32*(3), 293-324. https://doi.org/10.1216/jie.2020.32.293

#### Repository Citation

Guan, Wenchao; Jiang, Ying; and Xu, Yuesheng, "Computing the Newton Potential in the Boundary Integral Equation for the Dirichlet Problem of the Poisson Equation" (2020). *Mathematics & Statistics Faculty Publications*. 180.

https://digitalcommons.odu.edu/mathstat_fac_pubs/180

## Comments

Β© 2020 Rocky Mountain Mathematics Consortium.

Included with the kind written permission of the publisher.