Analysing Ewald%26apos;s method for the evaluation of Green%26apos;s functions for periodic media

作者：Arens Tilo^*; Sandfort Kai; Schmitt Susanne

来源：IMA Journal of Applied Mathematics, 2013, 78(3): 405-431.

DOI：10.1093/imamat/hxr057

• Green%26apos
• s function
• periodic media
• Helmholtz equation
• Ewald%26apos
• s method

Ewald%26apos;s method is a technique for efficiently evaluating periodic Green%26apos;s functions that is frequently used in the physics and engineering literature. The choice of the truncation and control parameters of the method is usually achieved by heuristic rules, and a rigorous numerical analysis of a fully discrete version of the method seems to be missing. In this paper, we analyse the truncation error of the method for the 2D and 3D periodic and biperiodic Green%26apos;s functions of the Helmholtz equation, respectively, providing new, explicit and sharp bounds. These estimates are subsequently used to study the effect of choosing the method%26apos;s control parameter and we give some recommendations for its choice. We present various numerical examples for the resulting method for evaluating the Green%26apos;s functions. The results are also carried over to evaluating the partial derivatives.

• 出版日期2013-6