The time harmonic Maxwell (THM) equations are of great interest in applied math- ematics and current physics applications, e.g., the excellence cluster PhoenixD. However, the numerical solution is challenging. This is specif- ically true for high wave numbers. Various solvers and preconditioners have been proposed, while the most promising are based on domain decomposition methods (DDM).

The goal of this work is to employ a domain decomposition method and to re-implement the algorithm in the modern finite element library deal.II. Therein, the construction of the subdomain interface conditions is a crucial aspect for which we use Impedance Boundary Conditions. Instead of handling the resulting linear system with a direct solver, which is typically done for the THM, we apply a well chosen block preconditioner to the linear system so we can solve it with an iterative solver like GMRES (generalized minimal residuals). Additionally high polynomial Nédélec elements are used in the implementation of the DDM. This implementation is computationally compared to several other (classical) preconditioners such as incomplete LU, additive Schwarz, Schur complement. These comparisons are done for different wave numbers. Higher wave numbers are well- known to cause challenges for numerical solution.

The work is done by S. Beuchler, S. Kinnewig, and T. Wick within the team led by  Prof. Thomas Wick

The original publication is available at https://link.springer.com/chapter/10.1007/978-3-030-95025-5_71

Read more and download the code from here:  
https://github.com/tommeswick
https://github.com/kinnewig