Parallel domain decomposition solvers for the time harmonic Maxwell equations

TL;DR

This paper develops and compares parallel finite element solvers for the time harmonic Maxwell equations, demonstrating that specialized block preconditioners combined with domain decomposition outperform standard methods, especially at high wave numbers.

Contribution

The paper introduces a novel combination of block preconditioners and domain decomposition methods tailored for high-frequency Maxwell problems, improving convergence and efficiency.

Findings

Standard preconditioners like ILU and Schwarz are slow for high wave numbers.

Schur complement methods also show slow convergence at high frequencies.

The proposed block preconditioner with DDM performs well for high wave number problems.

Abstract

The time harmonic Maxwell equations are of current interest in computational science and applied mathematics with many applications in modern physics. In this work, we present parallel finite element solver for the time harmonic Maxwell equations and compare different preconditioners. We show numerically that standard preconditioners like incomplete LU and the additive Schwarz method lead to slow convergence for iterative solvers like generalized minimal residuals, especially for high wave numbers. Even more we show that also more specialized methods like the Schur complement method also yield slow convergence. As an example for a highly adapted solver for the time harmonic Maxwell equations we use a combination of a block preconditioner and a domain decomposition method (DDM), which also preforms well for high wave numbers. Additionally we discuss briefly further approaches to solve high frequency problems more efficiently. Our developments are done in the open-source finite element library deal.II.