A Scalable Nonoverlapping and Nonconformal Domain Decomposition Method for Solving Time-Harmonic Maxwell Equations in $\mathbb{R}^3$

Abstract

We present a novel nonoverlapping and nonconformal domain decomposition method (DDM) for solving the time-harmonic Maxwell equations in $\mathbb{R}^3$. There are three major technical ingredients in the proposed nonconformal DDM: (a) a true second order transmission condition (SOTC) to enforce fields continuities across domain interfaces; (b) a corner edge penalty term to account for corner edges between neighboring subdomains; and (c) a global plane wave deflation technique to further improve the convergence of DDM for electrically large problems. It has been shown previously that a SOTC, which involves two second order transverse derivatives, facilitates convergence in the conformal domain decomposition method for both propagating and evanescent electromagnetic waves across domain interfaces. However, the discontinuous nature of the cement variables across the corner edges between neighboring subdomains remains troublesome. To mitigate the technical difficulty encountered and to enforce the needed divergence-free condition, we introduced a corner edge penalty term into the interior penalty formulation for the nonconformal DDM. The introduction of the corner edge penalty term successfully restored the superior performance of the SOTC. Finally, through an analysis of the DDM with the SOTC, we show that there still exists a weakly convergent region where the convergence in the DDM can still be unbearably slow for electrically large problems. Furthermore, it is found that the weakly convergent region is centered at the cutoff modes, or electromagnetic waves propagate in parallel to the domain interfaces. Subsequently, a global plane wave deflation technique is utilized to derive an effective global-coarse-grid preconditioner to promote fast convergence of the cutoff or near cutoff modes in the vicinity of domain interfaces. Finally, the strength of the proposed method is illustrated by means of three numerical examples.