Analysing Ewald's Method for the Evaluation of Green's Functions for Periodic Media

Expressions for periodic Green's functions for the Helmholtz equation in two and three dimensions are derived via Ewald's method. The decay rate of the series occurring in these expressions is analysed and rigorous estimates for the remainder are derived when the series are truncated and replaced by finite sums. The effect of choosing a control parameter occurring in Ewald's expressions is discussed and some recommendations for its choice are given based on the aforementioned estimates. We present various numerical examples for the resulting method for evaluating the Green's functions. The results can also be carried over to evaluating the partial derivatives.