Optimizing transmission conditions for multiple subdomains in the Magnetotelluric Approximation of Maxwell's equations

TL;DR

This paper investigates whether simplified two-subdomain optimized transmission conditions are sufficient for complex multi-subdomain decompositions in magnetotelluric approximations of Maxwell's equations, demonstrating their effectiveness across various configurations.

Contribution

It shows that optimizing transmission conditions in a two-subdomain setting is sufficient for more complex decompositions in diffusive Maxwell's equation approximations.

Findings

Asymptotically optimal transmission conditions are consistent across different subdomain counts.

Optimizing for two subdomains generalizes well to multiple subdomains in diffusion problems.

The same asymptotic behavior applies to large numbers of subdomains using limiting spectra.

Abstract

Classically transmission conditions between subdomains are optimized for a simplified two subdomain decomposition to obtain optimized Schwarz methods for many subdomains. We investigate here if such a simplified optimization suffices for the magnetotelluric approximation of Maxwell's equation which leads to a complex diffusion problem. We start with a direct analysis for 2 and 3 subdomains, and present asymptotically optimized transmission conditions in each case. We then optimize transmission conditions numerically for 4, 5 and 6 subdomains and observe the same asymptotic behavior of optimized transmission conditions. We finally use the technique of limiting spectra to optimize for a very large number of subdomains in a strip decomposition. Our analysis shows that the asymptotically best choice of transmission conditions is the same in all these situations, only the constants differ slightly. It is therefore enough for such diffusive type approximations of Maxwell's equations, which include the special case of the Laplace and screened Laplace equation, to optimize transmission parameters in the simplified two subdomain decomposition setting to obtain good transmission conditions for optimized Schwarz methods for more general decompositions.