Determine the optimal coupling parameter

Determine the theoretical dependence of convergence on the coupling parameter \(\alpha\) in the preconditioned \((n,t)\)-CFIE\(_\alpha\), and identify an optimal \(\alpha\) as a function of the Lamé parameters \((\lambda_j,\mu_j)\), density ratio \(\rho\), and frequency \(\omega\).

Background

The paper introduces a one-parameter family of indirect combined-field integral equations, denoted $(n,t)$-CFIE$_\alpha$, and proves unique solvability for every nonzero coupling parameter $\alpha$. Numerical experiments nevertheless show a substantial dependence of GMRES convergence on $\alpha$, particularly at high frequency: negative values, especially $\alpha=-1/2$, generally require fewer iterations than positive values. The paper does not provide a theoretical explanation for this dependence or a parameter-selection rule that accounts for the material parameters, density contrast, and frequency.