Green’s Functions for the Helmholtz Equation

2015 ◽  
pp. 397-588
Keyword(s):  
Helmholtz Equation ◽  
Green's Functions ◽  
Green’S Functions

scholarly journals On the unique solvability of a class of modified boundary integral equations

1984 ◽  
Vol 27 (3) ◽  
pp. 303-311 ◽  
Author(s):  
R. E. Kleinman ◽  
G. F. Roach
Keyword(s):  
Helmholtz Equation ◽  
Integral Equations ◽  
Unique Solvability ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Transmission Problem

In a recent paper the authors considered the transmission problem for the Helmholtz equation by using a reformulation of the problem in terms of a pair of coupled boundary integral equations with modified Green's functions as kernels. In this note we settle the question of the unique solvability of these modified boundary integral equations.

Fast algorithms for Helmholtz Green's functions

2008 ◽  
Vol 464 (2100) ◽  
pp. 3301-3326 ◽  
Author(s):  
Gregory Beylkin ◽  
Christopher Kurcz ◽  
Lucas Monzón
Keyword(s):  
Helmholtz Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Fast Algorithm ◽  
Green's Functions ◽  
Green’S Functions ◽  
Mixed Boundary ◽  
Spatial Domain ◽  
Fourier Domain ◽  
Computational Domain

The formal representation of the quasi-periodic Helmholtz Green's function obtained by the method of images is only conditionally convergent and, thus, requires an appropriate summation convention for its evaluation. Instead of using this formal sum, we derive a candidate Green's function as a sum of two rapidly convergent series, one to be applied in the spatial domain and the other in the Fourier domain (as in Ewald's method). We prove that this representation of Green's function satisfies the Helmholtz equation with the quasi-periodic condition and, furthermore, leads to a fast algorithm for its application as an operator. We approximate the spatial series by a short sum of separable functions given by Gaussians in each variable. For the series in the Fourier domain, we exploit the exponential decay of its terms to truncate it. We use fast and accurate algorithms for convolving functions with this approximation of the quasi-periodic Green's function. The resulting method yields a fast solver for the Helmholtz equation with the quasi-periodic boundary condition. The algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when Green's function is applied to discontinuous functions or potentials with singularities. We also construct Helmholtz Green's functions with Dirichlet, Neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi-periodic Green's function to apply them. The complexity, in dimension d ≥2, of these algorithms is ( κ d  log  κ + C (log  ϵ −1 ) d ), where ϵ is the desired accuracy, κ is proportional to the number of wavelengths contained in the computational domain and C is a constant. We illustrate our approach with examples.

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