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.


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