# Periodic Green's functions in integral equation methods in different frequency regimes

I'm asking about the solution of the Helmholtz equation on a periodic domain with piecewise constant wavespeed in different frequency regimes. One possible approach is to solving this problem is to write down integral equations on the boundary surfaces in terms of the Green's function of the system. Since the domain is periodic, this will be a periodic Green's function like $$G(r,r') = \sum_L G_0(r,r'+L)$$ where $L$ is a lattice vector, and $G_0$ is something like $$G_0(r,r') = \frac{e^{ik(r-r')}}{|r-r'|}$$ My question concerns how the computational cost of this method scales with frequency ($k$). Does it become computationally more difficult at low or high frequencies due to needing to include more terms in the lattice sum?

Edit: People seem to be answering different questions than the one I'm asking. I should clarify that I'm not interested in implementing such a method. I'm merely asking about the theoretical difficulties, as background for understanding the strengths and (moreso) weakness of the method in different frequency regimes. The kinds of problems I have in mind are (more or less) computing modes of periodic arrays of waveguides.

## 2 Answers

The decay properties of the Green's function depend, among other things, on the coefficients in your equations. For example, lower-dimensional wave guides typically transport information for long distances, and you may have to add up a lot of terms for good accuracy.

Alternatives are to write the Green's function as a sum of sines and cosines that already satisfy the periodic boundary conditions, or as a sum of eigenfunctions of the operator.

Though I have never implemented this myself, the conventional wisdom I've heard is that the spatial summation converges very slowly and it's preferable to use the Ewald transform, e.g. http://w3.uniroma1.it/lovat/giampiero/Documentipdf/IJ24.pdf

There's probably boatloads of papers on this subject in IEEE transactions.