Fast potential theory. II: Layer potentials and discrete sums. (English) Zbl 0753.65088
From author’s abstract: Three new families of fast algorithms for classical potential theory, based on Ewald summation and fast transforms of Gaussians and Fourier series are presented.
Ewald summation separates the Green function for a cube into a high- frequency localized part and a rapidly-converging Fourier series. Each part can then be evaluated efficiently with appropriate fast transform algorithms. Our algorithms are naturally suited to the use of Green functions with boundary conditions imposed on the boundary of a cube, rather than free-space Green functions.
Two-dimensional numerical results are presented for all algorithms. Layer potentials are evaluated to second-order accuracy, in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation.
Ewald summation separates the Green function for a cube into a high- frequency localized part and a rapidly-converging Fourier series. Each part can then be evaluated efficiently with appropriate fast transform algorithms. Our algorithms are naturally suited to the use of Green functions with boundary conditions imposed on the boundary of a cube, rather than free-space Green functions.
Two-dimensional numerical results are presented for all algorithms. Layer potentials are evaluated to second-order accuracy, in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation.
Reviewer: U.Göhner (Stuttgart)
MSC:
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |
| 65R20 | Numerical methods for integral equations |
| 35C15 | Integral representations of solutions to PDEs |
| 31A30 | Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
discrete sums; Poisson equation; fast algorithms; potential theory; Ewald summation; fast transforms of Gaussians; Fourier series; Green function; numerical results; Layer potentialsCitations:
Zbl 0685.65109References:
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