True 2~(nd) Order Transmission Condition in Conjunction with Corner Edge Penalty Term for Non-conformal Domain Decomposition Methods in Solving Time-Harmonic Maxwell Equations

摘要

Non-overlapping domain decomposition methods (DDMs) have been shown to provide efficient iterative algorithm for the finite element (FE) solution of the time-harmonic electromagnetic wave problems [1-4]. It is well known that the convergence behavior of non-overlapping DD methods is directly related to the transmission conditions (TCs) used to enforce the continuity of tangential fields on the interface between sub-domains. Most often, a 1st order complex Robin TC (FOTC) is used. However, by including higher order derivatives in the transverse direction, the convergence of iterative algorithms can be improved. In [3], a new type of SOTC, called SOTC-TE, is shown to considerably improve the convergence w.r.t. FOTC. But, it is only effective in preconditioning one set of problematic eigenvalues. The eigenmodes neglected by the SOTC-TE, namely the transverse magnetic (TM) evanescent modes, present the last impediment to solver convergence. We address these modes by introducing a full second-order TC (SOTC-Full) that includes an additional term with a second-order transverse derivative. An analysis using a simplified problem shows that, the SOTC-Full shifts both TE and TM evanescent eigen-values away from the origin, and does not alter the convergence of the propagating modes when compared to the FOTC. However, when the SOTC-Full is applied to non-conformal DDMs, it is quickly discovered that the performance does not achieve the expected speed-up as in the conformal DDMs. We have found that the root cause of such a defect, it is mainly due to the enlargement of the function space for the auxiliary cement variables on the interfaces, namely the use of the discontinuous curl-conforming basis functions for the auxiliary variables. Consequently, the non-conformal DDMs allow for eigen-modes, whose magnetic fluxes do not satisfy the needed divergence-free condition on the corner edges. To mitigate such a malady, we employ the interior penalty formulation and introduce additional corner penalty term relating to the divergence free constraint for the cement variables. The introduction of the corner edge penalty terms in the IP formulation restores the full benefits of the 2nd order TC in the non-conformal DDMs.