Optimized Schwarz Methods for the Time-Harmonic Maxwell Equations with Damping

Abstract

In a previous paper, two of the authors have proposed and analyzed an entire hierarchy of optimized Schwarz methods for Maxwell's equations both in the time-harmonic and time-domain case. The optimization process has been perfomed in a particular situation where the electric conductivity was neglected. Here, we take into account this physical parameter which leads to a fundamentally different analysis and a new class of algorithms for this more general case. From the mathematical point of view, the approach is different, since the algorithm does not encounter the pathological situations of the zero-conductivity case and thus the optimization problems are different. We analyze one of the algorithms in this class in detail and provide asymptotic results for the remaining ones. We illustrate our analysis with numerical results.

Schwarz Methods for Second Order Maxwell Equations in 3D with Coefficient Jumps

“…If Z = Y,Y −1 and Y = 1, the optimized convergence factor ρ * opt in (12) has the asymptotic behavior: . Then we obtain ρ Mopt ≥ ρ Eopt and thus (18).…”

“…Optimized Schwarz methods have been developed for Maxwell equations in first order form without conductivity in [8], and with conductivity in [5,12]. These methods use modified transmission conditions, and often converge much faster than classical Schwarz methods.…”

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].…”

Optimized Schwarz Methods for Circular Domain Decompositions with Overlap

“…Among the many domain decomposition methods, optimized Schwarz methods, going back to an idea for a nonoverlapping method by Lions [28], have attracted substantial attention over the past decade because their transmission conditions can be adapted to the physics of the underlying problem and thus lead to very efficient methods for hard problems; for an overview and references, see [35,17]. Optimized Schwarz methods are a very active area of research, and they have found their way into many areas of applications: for Helmholtz problems, see [12,6,22], for advection diffusion evolution problems, see [34,20,7,40], for wave equations, see [21,19], for Maxwell problems, see [41,13,38,37,8], and for shallow water and the primitive equations, see [39,3]. Optimized Schwarz methods have also led to new theoretical developments (see, for example, [27,30,29]) and many innovative preconditioners that appeared under different names; see, for example, the sweeping preconditioner in [16,2,23,14], the source transfer method in [11], and also [42], which are among the best iterative…”