A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition

Li, Yonglin

doi:10.4208/csiam-am.2020-0025

Cited by 16 publications

(4 citation statements)

References 27 publications

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“…(ii) The CIP-FEM was first introduced by Douglas and Dupont [21] for second-order elliptic and parabolic PDEs, and has been applied to the the Helmholtz problem (1.1) with the impedance, DtN, and PML boundary conditions; see, e.g., [22,46,47,55,57,59]. The CIP-FEM has been proved to be highly effective in reducing pollution errors, while the work [46] on PML boundary condition focused only on the linear CIP-FEM.…”

Section: The Cip-femmentioning

confidence: 99%

“…To our knowledge, this is the best result by far for the preasymptotic error estimate of higher-order FEM. Numerous approaches have emerged over the past two decades to reduce the pollution errors, including hp-FEM [9,40,50,51], CIP-FEM [22,46,47,55,59], discontinuous Galerkin method (DG) [23,24,32,49,58], Trefftz methods [30,31,[34][35][36][37][38]42], and multiscale methods [8,29,52]. In this paper, we would like to introduce the higher-order CIP-FEM which offers significant advantages in reducing pollution errors.…”

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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Section: The Cip-femmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…These volumetric discretization approaches can treat problems in general geometries and including spatially varying media. As is well known, however, these methods typically suffer from spatial and temporal numerical dispersion errors (also known as pollution errors [6,38]), and they therefore require use of fine spatial and temporal meshes-and thus, large computer-memory and run-times-to achieve accurate solutions in applications involving high frequencies and/or long time simulations.…”

Section: Introductionmentioning

confidence: 99%

Multiple-scattering frequency-time hybrid solver for the wave equation in interior domains

Bruno¹,

Yin²

2022

Preprint

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This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in twodimensional interior spatial domains. The approach relies on four main elements, namely, 1) A multiple scattering strategy that decomposes a given time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open-arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems; 2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point 1); 3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation; and, 4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem-which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology.

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“…The numerical results show that the adaptive finite element DtN method is competitive with the adaptive finite element PML method. We refer to [27] for a continuous interior penalty finite element method (CIP-FEM) for solving high frequency scattering problems with the truncated DtN boundary condition.…”

mentioning

confidence: 99%

An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

Bao¹,

Zhang²,

Hu³

et al. 2021

Discrete &Amp; Continuous Dynamical Systems - B

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This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

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A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition

Cited by 16 publications

References 27 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

Multiple-scattering frequency-time hybrid solver for the wave equation in interior domains

An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

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