Local Multiple Traces Formulation for Heterogeneous Electromagnetic Scattering: Implementation and Preconditioning

TL;DR

This paper introduces a scalable local multiple traces formulation for 3D electromagnetic scattering involving heterogeneous media, improving implementation and preconditioning strategies for complex domain decompositions.

Contribution

It presents a novel framework using skeleton meshes for efficient EM MTF implementation and explores advanced preconditioning techniques for better solver performance.

Findings

Effective handling of shared interfaces among subdomains.

Enhanced scalability with skeleton mesh-based implementation.

Improved convergence with block OSRC preconditioning.

Abstract

We consider the three-dimensional time-harmonic electromagnetic (EM) wave scattering transmission problem involving heterogeneous scatterers. The fields are approximated using the local multiple traces formulation (MTF), originally introduced for acoustic scattering. This scheme assigns independent boundary unknowns to each subdomain and weakly enforces Calder\'on identities along with interface transmission conditions. As a result, the MTF effectively handles shared points or edges among multiple subdomains, while supporting various preconditioning and parallelization strategies. Nevertheless, implementing standard solvers presents significant challenges, particularly in managing the degrees of freedom associated with subdomains and their interfaces. To address these difficulties, we propose a novel framework that suitably defines approximation spaces and enables the efficient exchange of normal vectors across subdomain boundaries. This framework leverages the skeleton mesh, representing the union of all interfaces, as the computational backbone, and constitutes the first scalable implementation of the EM MTF. Furthermore, we conduct several numerical experiments, exploring the effects of increasing subdomains and block On-Surface-Raditation-Condition (OSRC) preconditioning, to validate our approach and provide insights for future developments.