A new addition theorem for the 3-D Navier-Lam\'e system and its application to the method of fundamental solutions

TL;DR

This paper introduces a new addition theorem for the fundamental solution of the 3-D Navier-Lamé system, enhancing numerical methods like boundary element and fundamental solutions methods.

Contribution

It provides an explicit expansion involving Bessel functions and spherical harmonics, improving efficiency in exterior domain approximations.

Findings

Efficient fundamental solution expansion using Bessel functions and spherical harmonics.

Enhanced boundary element and fundamental solutions methods for Navier-Lamé system.

Validated effectiveness in exterior domain problems.

Abstract

We obtain a new addition theorem for the fundamental solution of the Navier-Lam\'e system in dimension 3 satisfying the Kupradze radiation conditions. This provides an expansion of this fundamental solution that involves only the evaluation of Bessel functions and scalar spherical harmonics. This is particularly useful in collocation numerical methods based on fundamental solutions, such as the boundary element method or the method of fundamental solutions. For this last method, we show its efficiency when approximating the Navier-Lam\'e system in exterior domains.