Parallel domain decomposition solvers for the time harmonic Maxwell equations

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.