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

Description

Classical Schwarz methods need in general overlap to converge, but in the case of hyperbolic problems, they can also be convergent without overlap, see [7]. For the first order formulation of Maxwell equations, we have proved however in [18] that the classical Schwarz method without overlap does not converge in most cases in the presence of coefficient jumps aligned with interfaces. Optimized Schwarz methods have been developed for Maxwell equations in first order form without conductivity in [8], and with conductivity in [5, 12]. These meth- ods use modified transmission conditions, and often converge much faster than clas- sical Schwarz methods. For DG discretizations of Maxwell equations, optimized Schwarz methods can be found in [9, 10, 6]. Optimized Schwarz methods were also developed for the second order formulation of Maxwell equations, see [1], and [16, 17] for scattering problems with applications. While usually coefficient jumps hamper the convergence of domain decomposi- tion methods, this is very different for optimized Schwarz methods. For diffusive problems, it was shown in [11] that jumps in the coefficients can actually lead to faster iterations, when they are taken into account correctly in the transmission con- ditions: optimized Schwarz methods benefit from jumps in the coefficients at inter- faces. We had shown in [18] that this also holds for the special case of transverse magnetic modes (TMz) in the two dimensional first order Maxwell equations. We show in this short paper that these results for the TMz modes (and the correspond- ing ones for the transverse electric modes (TEz)) can be used to formulate optimized Schwarz methods for the 3D second order Maxwell equations which then in some cases converge faster, the bigger the coefficient jumps are.

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