# Scalable solvers for complex electromagnetics problems

## TL;DR

This paper introduces scalable domain decomposition methods for complex 3D electromagnetics problems that are robust to material heterogeneity and coefficient jumps, without requiring spectral information.

## Contribution

The work presents a novel, scalable preconditioning approach for edge finite element discretizations that handles arbitrary heterogeneity and material interfaces efficiently.

## Key findings

- Method is robust to coefficient jumps and heterogeneity.
- Achieves excellent weak scalability in 3D realistic cases.
- Does not require spectral information unlike existing methods.

## Abstract

In this work, we present scalable balancing domain decomposition by constraints methods for linear systems arising from arbitrary order edge finite element discretizations of multi-material and heterogeneous 3D problems. In order to enforce the continuity across subdomains of the method, we use a partition of the interface objects (edges and faces) into sub-objects determined by the variation of the physical coefficients of the problem. For multi-material problems, a constant coefficient condition is enough to define this sub-partition of the objects. For arbitrarily heterogeneous problems, a relaxed version of the method is defined, where we only require that the maximal contrast of the physical coefficient in each object is smaller than a predefined threshold. Besides, the addition of perturbation terms to the preconditioner is empirically shown to be effective in order to deal with the case where the two coefficients of the model problem jump simultaneously across the interface. The new method, in contrast to existing approaches for problems in curl-conforming spaces does not require spectral information whilst providing robustness with regard to coefficient jumps and heterogeneous materials. A detailed set of numerical experiments, which includes the application of the preconditioner to 3D realistic cases, shows excellent weak scalability properties of the implementation of the proposed algorithms.