Numerical approximation of the scattering amplitude in elasticity

TL;DR

This paper introduces a numerical method to approximate scattering amplitudes in elastic media with variable potentials, using adapted Lippmann-Schwinger equations, with proven convergence and demonstrated through numerical examples.

Contribution

It adapts Vainikko's method to solve Lippmann-Schwinger equations for the Lamé operator in elasticity, providing convergence analysis and implementation details.

Findings

Method successfully approximates scattering amplitudes

Convergence proven for smooth potentials

Numerical examples validate the approach

Abstract

We propose a numerical method to approximate the scattering amplitudes for the elasticity system with a non-constant matrix potential in dimensions $d=2$ and $3$. This requires to approximate first the scattering field, for some incident waves, which can be written as the solution of a suitable Lippmann-Schwinger equation. In this work we adapt the method introduced by G. Vainikko in \cite{V} to solve such equations when considering the Lam\'e operator. Convergence is proved for sufficiently smooth potentials. Implementation details and numerical examples are also given.