Optimized Schwarz Methods for Curl-Curl Time-Harmonic Maxwell’s Equations

Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl–curl Maxwell's equations

“…The results for (12) can be obtained similarly: we get −λL is of the same form shown in (37). One only has to take the coefficients α,α, β,β from (36) for the algorithm using transmission conditions (11), and from (39) for the algorithms using transmission conditions (12).…”

“…for more details, see [11]. Using the TE-TM decomposition, we show in the next section that the two general forms of transmission conditions (11) and (12) lead apart from a few subtleties to optimized Schwarz methods with the same contraction factor, and therefore the complete optimization results obtained in [12] for the first order formulation (2) can be used for all these families of Schwarz methods for the curl-curl formulation (3) to obtain the fastest methods in each class.…”

“…In order to show the relationship between the to families of transmission conditions (11) and (12), we use Fourier analysis, and thus assume that the coefficients are constant, and the domain on which the original problem is posed is Ω = R 3 , in which case we need to impose the Silver-Müller radiation condition lim r→∞ r (∇ × E × n + iωE) = 0, where r = |x|, n = x/|x|, for the problem to be well-posed [31]. The two subdomains…”

Optimized Schwarz Domain Decomposition Methods for Scalar and Vector Helmholtz Equations

“…The first developments in this direction were introduced by Després [13,14], who used simple impedance boundary conditions on the interfaces. A great variety of more general impedance conditions has been proposed since these early works, leading to so-called optimized Schwarz domain decomposition methods for time-harmonic wave problems [1,9,10,11,14,15,16,19,20,25,27,43,44,45]. These methods can be used with or without overlap between the subdomains, and their convergence rate strongly depends on the transmission condition.…”

“…However, using the DtN leads to a very expensive numerical procedure in practice, as this operator is non-local. Practical algorithms are thus based on local approximations of these operators, both for the acoustic case [13,9,10,11,27] and the electromagnetic one [1,14,15,16,20,21,43,44,45]. Recently, PMLs have also been used for this same purpose [23,47,51,52].…”