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Lattice sums and the two-dimensional, periodic Green’s function for the Helmholtz equation. (English) Zbl 1048.78015

Summary: Many algorithms that are currently used for the solution of the Helmholtz equation in periodic domains require the evaluation of the Green’s function \(G(x,x_0)\). The fact that the natural representation of \(G\) via the method of images gives rise to a conditionally convergent series whose direct evaluation is prohibitive has inspired the search for more efficient procedures for evaluating this Green’s function. Recently, the evaluation of \(G\) through the ‘lattice-sum’ representation has proven to be both accurate and fast. As a consequence, the computation of the requisite, also conditionally convergent, lattice sums has become an active area of research. We describe a new integral representation for these sums, and compare our results with other techniques for evaluating similar quantities.

MSC:

78M25 Numerical methods in optics (MSC2010)
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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