Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem

Bramble, James H.; Pasciak, Joseph E.

doi:10.1090/s0025-5718-07-02037-6

Cited by 55 publications

(50 citation statements)

References 15 publications

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“…The uniqueness of solutions to the PML problem (2.7) (for k real and positive) now follows from the above theorem and the proof of Theorem 1 of [11]. Following [9], the perturbed form…”

Section: If and Only If The Seriesmentioning

confidence: 89%

“…The PML approach [9] provides a convenient way to deal with (2.1) with the outgoing condition. We will use a spherical PML (see Figure 1).…”

Section: Pml and The Resonance Problemmentioning

confidence: 99%

“…The PML problem corresponding to a scattering problem was studied in [9], and the techniques there easily extend to our problem. The uniqueness of solutions to the PML problem (2.7) (for k real and positive) now follows from the above theorem and the proof of Theorem 1 of [11].…”

Section: If and Only If The Seriesmentioning

confidence: 99%

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The computation of resonances in open systems using a perfectly matched layer

Kim

Pasciak

2009

Math. Comp.

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Abstract. In this paper, we consider the problem of computing resonances in open systems. We first characterize resonances in terms of (improper) eigenfunctions of the Helmholtz operator on an unbounded domain. The perfectly matched layer (PML) technique has been successfully applied to the computation of scattering problems. We shall see that the application of PML converts the resonance problem to a standard eigenvalue problem (still on an infinite domain). This new eigenvalue problem involves an operator which resembles the original Helmholtz equation transformed by a complex shift in the coordinate system. Our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite (computational) domain with a convenient boundary condition and second by applying finite elements on the computational domain. We shall prove that the first of these steps leads to eigenvalue convergence (to the desired resonance values) which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large. The analysis of the second step is classical. Finally, we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions.

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“…The uniqueness of solutions to the PML problem (2.7) (for k real and positive) now follows from the above theorem and the proof of Theorem 1 of [11]. Following [9], the perturbed form…”

Section: If and Only If The Seriesmentioning

confidence: 89%

“…The PML approach [9] provides a convenient way to deal with (2.1) with the outgoing condition. We will use a spherical PML (see Figure 1).…”

Section: Pml and The Resonance Problemmentioning

confidence: 99%

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The computation of resonances in open systems using a perfectly matched layer

Kim

Pasciak

2009

Math. Comp.

Self Cite

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“…These two formulations are also mathematically identical, and the fact was proved by Zhao and Cangellaris, provided that the electric and magnetic fields presented in the Chew-Weedon stretched-coordinate formulation are properly defined [9]. Bramble and Pasciak have shown the existence and uniqueness of solutions to the truncated time harmonic PML problem provided that the truncated domain is sufficiently large [10]. Bao and Wu have given the convergence analysis in spherical coordinates for a three-dimensional electromagnetic scattering problem and established an explicit error estimate between the solution of the scattering problem and that of the truncated PML problem [11].…”

Section: Introductionmentioning

confidence: 90%

Stability Analysis of Yee Schemes to PML and UPML for Maxwell Equations

Ying¹

2009

JCM

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We utilize Fourier methods to analyze the stability of the Yee difference schemes for Bérenger PML (perfectly matched layer) as well as the UPML (uniaxial perfectly matched layer) systems of two-dimensional Maxwell equations. Using a practical spectrum stability concept, we find that the two schemes are spectrum stable under the same conditions for mesh sizes. Besides, we prove that the UPML schemes with the same damping in both directions are stable. Numerical examples are given to confirm the stability analysis for the PML method.Mathematics subject classification: 65M12.

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“…Refs. [16,17]), causality [18,19], long-time behavior [16,20], termination of the PML [21,22], and finite-element dispersion and convergence analysis in the frequency domain [23,24]. A recommendable review of the PML in the context of Maxwell's equations is presented in Ref.…”

Section: Introductionmentioning

confidence: 99%

Improving the performance of perfectly matched layers by means of hp‐adaptivity

Michler

Demkowicz

Kurtz

et al. 2007

Numerical Methods Partial

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We improve the performance of the Perfectly Matched Layer (PML) by using an automatic hp-adaptive discretization. By means of hp-adaptivity, we obtain a sequence of discrete solutions that converges exponentially to the continuum solution. Asymptotically, we thus recover the property of the PML of having a zero reflection coefficient for all angles of incidence and all frequencies on the continuum level. This allows us to minimize the reflections from the discrete PML to an arbitrary level of accuracy while retaining optimal computational efficiency. Since our hp-adaptive scheme is automatic, no interaction with the user is required. This renders tedious parameter tuning of the PML obsolete. We demonstrate the improvement of the PML performance by hp-adaptivity through numerical results for acoustic, elastodynamic, and electromagnetic wave-propagation problems in the frequency domain and in different systems of coordinates.

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Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem

Cited by 55 publications

References 15 publications

The computation of resonances in open systems using a perfectly matched layer

The computation of resonances in open systems using a perfectly matched layer

Stability Analysis of Yee Schemes to PML and UPML for Maxwell Equations

Improving the performance of perfectly matched layers by means of hp‐adaptivity

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