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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.

78M25Numerical methods in optics
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
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