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

Dolean, Victorita; Gander, Martin J.; Veneros, Erwin

doi:10.1007/978-3-319-18827-0_48

Cited by 3 publications

(8 citation statements)

References 14 publications

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“…Different choices of S j , j = 1, 2 lead to different Schwarz methods. Since it was shown in [21] that it is sufficient to study the 2d TMz and TEz variants of (2) to get a complete understanding of (2), we focus in what follows on the 2d case.…”

Section: Schwarz Methods For Maxwell's Equationsmentioning

confidence: 99%

“…We now study transmission conditions for the TEz mode for the physically important case µ 1 = µ 2 and ε 1 = ε 2 . Using Remark 1, one can also obtain an equivalent result for the TEz mode for the case µ 1 = µ 2 and ε 1 = ε 2 , which was announced without proof and in less general form in [20], and used in [21] to obtain results for the 3d case.…”

Section: Transmission Conditions For the Tez Modementioning

confidence: 97%

“…The case ω 1 = ω 2 is a resonance case that was studied in [22] and needs special treatment, like the particular resonance case where ε and µ are continuous, see [8]. The fact to have a jump in ε helps convergence, and Theorem 3 below was presented in a less general form in [20] without proof, and was the main building block to understand the 3d case in [21]. We give now the general result, and a detailed proof based on asymptotic analysis.…”

Section: Transmission Conditions For the Tmz Modementioning

confidence: 99%

“…For a special case in 2d, a convergence result stated without proof in [20] showed that well chosen transmission conditions can lead to non-overlapping Schwarz algorithms that converge independently of the mesh parameter. In addition it was shown in [21] that complete results for the 2d transverse magnetic (TMz) and transverse electric (TEz) modes 1 will imply directly convergence results in 3d as well. The purpose of this manuscript is to prove the 2d results announced in [20] using asymptotic analysis.…”

Section: Introductionmentioning

confidence: 98%

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Asymptotic analysis of optimized Schwarz methods for maxwell’s equations with discontinuous coefficients

2018

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Discretized time harmonic Maxwell’s equations are hard to solve by iterative methods, and the best currently available methods are based on domain decomposition and optimized transmission conditions. Optimized Schwarz methods were the first ones to use such transmission conditions, and this approach turned out to be so fundamentally important that it has been rediscovered over the last years under the name sweeping preconditioners, source transfer, single layer potential method and the method of polarized traces. We show here how one can optimize transmission conditions to take benefit from the jumps in the coefficients of the problem, when they are aligned with the subdomain interface, and obtain methods which converge for two subdomains in certain situations independently of the mesh size, which would not be possible without jumps in the coefficients.

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Section: Schwarz Methods For Maxwell's Equationsmentioning

confidence: 99%

Section: Transmission Conditions For the Tez Modementioning

confidence: 97%

Section: Transmission Conditions For the Tmz Modementioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Asymptotic analysis of optimized Schwarz methods for maxwell’s equations with discontinuous coefficients

2018

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“…This enables one to solve Maxwell's equations in the strong form on each of the sub-domains, which is the route that we will pursue in this paper. For a recent exposition on a deterministic domain decomposition method for the three-dimensional time-dependent Maxwell's equations, see [10].…”

Section: Introductionmentioning

confidence: 99%

Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem

et al. 2016

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Stochastic domain decomposition is proposed as a novel method for solving the two-dimensional Maxwell's equations as used in the magnetotelluric method. The stochastic form of the exact solution of Maxwell's equations is evaluated using Monte-Carlo methods taking into consideration that the domain may be divided into neighboring sub-domains. These sub-domains can be naturally chosen by splitting the sub-surface domain into regions of constant (or at least continuous) conductivity. The solution over each sub-domain is obtained by solving Maxwell's equations in the strong form. The sub-domain solver used for this purpose is a meshless method resting on radial basis function based finite differences. The method is demonstrated by solving a number of classical magnetotelluric problems, including the quarter-space problem, the block-in-half-space problem and the triangle-in-half-space problem.

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