Wavenumber explicit analysis for time-harmonic Maxwell equations in a spherical shell and spectral approximations

Ma, Lina; Shen, Jie; Wang, Li-Lian; Yang, Zhiguo

doi:10.1093/imanum/drx014

Cited by 5 publications

(5 citation statements)

References 39 publications

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“…Similar to the 2D case, it suffices to prove that It remains to prove for ℓ = m = 0. From (2.30), 42) and similarly,…”

Section: Proof Of Theorem 31 For 3d Casementioning

confidence: 83%

“…Our key idea for proving (1.6) is to use the harmonic expansion of the truncated PML solution and analyze each term carefully in the expansion by using various properties of the Bessel functions (cf. [4,8,42]). The estimates of the inf-sup constant in (1.4) are proved by using (1.6) and following the proofs in [12].…”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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FEM and CIP-FEM for Helmholtz Equation with High Wave Number and Perfectly Matched Layer Truncation

Li¹,

Wu²

2019

SIAM J. Numer. Anal.

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The Helmholtz scattering problem with high wave number is truncated by the perfectly matched layer (PML) technique and then discretized by the linear continuous interior penalty finite element method (CIP-FEM). It is proved that the truncated PML problem satisfies the inf-sup condition with inf-sup constant of order O(k −1 ). Stability and convergence of the truncated PML problem are discussed. In particular, the convergence rate is twice of the previous result. The preasymptotic error estimates in the energy norm of the linear CIP-FEM as well as FEM are proved to be C1kh + C2k 3 h 2 under the mesh condition that k 3 h 2 is sufficiently small. Numerical tests are provided to illustrate the preasymptotic error estimates and show that the penalty parameter in the CIP-FEM may be tuned to reduce greatly the pollution error. R 0 J n (kt)f n (t)tdt.(2.24)Note that from [49, 10.21(i)], the zeros of J ν (z) are all real for ν ≥ −1. SinceR is not real and J −n = (−1) n J n for integer n, we have J n (kR) = 0, n ∈ Z.

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“…Similar to the 2D case, it suffices to prove that It remains to prove for ℓ = m = 0. From (2.30), 42) and similarly,…”

Section: Proof Of Theorem 31 For 3d Casementioning

confidence: 83%

Section: )mentioning

confidence: 99%

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FEM and CIP-FEM for Helmholtz Equation with High Wave Number and Perfectly Matched Layer Truncation

Li¹,

Wu²

2019

SIAM J. Numer. Anal.

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“…identities [68,26] (also see, e.g., [54,16,70]), which applies only to star-shaped domains (or the bounded scatterer is of star-shape, see [16]). For Maxwell's equations, Hiptmair et al [55] (and independently by [37]) derived for the first time the wavenumber explicit estimates for the time-harmonic Maxwell's equations but with an approximate boundary condition: (∇ × E) × e r − ikE T = h. Ma et al [66] considered the exact DtN boundary conditions for Maxwell's system in the spherical geometry.…”

Section: Time-domain Palmentioning

confidence: 99%

“…As already mentioned in Chapter 1, significant efforts have been devoted to the wavenumber explicit analysis for the Helmholtz equations with lower-order absorbing boundary conditions or exact Dirichlet-to-Neumann (DtN) transparent boundary conditions. We refer to the introductory sections of the recent papers [66,64] for review of many existing works along this line. However, such an analysis for the PML technique is very limited.…”

Section: Circular Palmentioning

confidence: 99%

Perfect absorbing layers for wave scattering problems and related topics

Gao¹

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List of Figures 2.3 Profiles of the PML solution U p with regular function (2.14) (σ 0 = 1, n = 2) and singular function (2.15) vs PAL solution U a in Ω and V a in Ω d with k = 99.99, (L, d) = (1, 0.2) along the x-axis at y = π/2. Left: real part solution; Right: imaginary part solution.. .. .. .. .. . 2.4 Left: illustration of the circular PAL from radial direction. Middle: schematic illustration of the rectangular PAL Ω d with four trapezoidal patches. Right: illustration of the elliptic PAL under elliptic coordinate. 2.5 Left: errors against various N for different layers (width d = 0.1) with boundary condition g(y) = sin(5y) − sin(30y); Right: errors against various N for different layers (width d = 0.5) with boundary condition g(y) = sin(5y) − sin(30y).

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Energy-Conserved Splitting Multidomain Legendre-Tau Spectral Method for Two Dimensional Maxwell’s Equations

Niu

Dong

2022

J Sci Comput

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Wavenumber explicit analysis for time-harmonic Maxwell equations in a spherical shell and spectral approximations

Cited by 5 publications

References 39 publications

FEM and CIP-FEM for Helmholtz Equation with High Wave Number and Perfectly Matched Layer Truncation

FEM and CIP-FEM for Helmholtz Equation with High Wave Number and Perfectly Matched Layer Truncation

Perfect absorbing layers for wave scattering problems and related topics

Energy-Conserved Splitting Multidomain Legendre-Tau Spectral Method for Two Dimensional Maxwell’s Equations

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