Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

Domínguez, Vı́ctor; Lyon, Mark; Turc, Catalin

doi:10.1216/jie-2016-28-3-395

Cited by 18 publications

(45 citation statements)

References 52 publications

(117 reference statements)

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“…Throughout this paper we utilize a high-order Nyström method for the spatial discretization of the windowed frequency-domain BIEs (16). Nyström methods enjoy well-known advantages over other boundary integral equation methods.…”

Section: Nyström Methods and Frequency-domain Problemsmentioning

confidence: 99%

“…The numerical errors in the fields are displayed in Figure 2a where it can be clearly seen that our solver yields the expected order of convergence. The numerical error is here defined as max{e (1) , e (2) } where A in (19) were obtained by numerically solving the windowed BIE (16) and then evaluating the representation formula (17), using the approximate surface densities. The reference fields (ũ (j) ) were produced using a fine grid consisting of 512 discretization points.…”

Section: Nyström Methods and Frequency-domain Problemsmentioning

confidence: 99%

“…In order to demonstrate the high-order convergence of the WGF method in the context of CQ methods, we next consider the frequency-domain problems of the previous example but they are now solved for various window sizes A > 0. The number of discretization points used in this example is such that ∼15 per unit length are used, which turns out to be enough to guarantee that the dominant error in all the calculations stems from the use of a finite window size A > 0 and not from the Nyström discretization of the windowed BIE (16). Figure 2b displays the numerical errors obtained for the various window sizes and complex wavenumbers considered.…”

Section: Nyström Methods and Frequency-domain Problemsmentioning

confidence: 99%

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Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

2019

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This paper presents a class of boundary integral equation methods for the numerical solution of acoustic and electromagnetic time-domain scattering problems in the presence of unbounded penetrable interfaces in two-spatial dimensions. The proposed methodology relies on Convolution Quadrature (CQ) methods in conjunction with the recently introduced Windowed Green Function (WGF) method. As in standard time-domain scattering from bounded obstacles, a CQ method of the user's choice is utilized to transform the problem into a finite number of (complex) frequency-domain problems posed on the domains involving penetrable unbounded interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF method-which introduces errors that decrease super-algebraically fast as the window size increases. The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off-the-shelf Helmholtz boundary integral equation solver capable of handling complex wavenumbers with a large imaginary part. A high-order Nyström method based on Alpert quadrature rules is utilized here. A variety of numerical examples including wave propagation in open waveguides as well as scattering from multiply layered media, demonstrate the capabilities of the proposed approach.(CQ) methods [25,26], in particular, have effectively enabled the use of (complex) frequencydomain boundary integral equation (BIE) solvers to tackle a variety of wave propagation problems, by providing a stable procedure to discretize the associated convolution equations for the unknown time evolution of the relevant surface densities; see [33] for the mathematical foundations of the method, and [5,19] for details on the algorithmic implementation. In the case of the scalar wave equation with piecewise constant wavespeed, to which this paper is devoted to, approximate traces at discrete times are produced all at once from a finite sequence of independent Helmholtz problems that can be solved in parallel by means of BIE methods. Although this CQ-BIE approach has proven to be competitive to volume discretization methods in the context of obstacle scattering problems [3,4,34], its extension to problems involving unbounded material interfaces is severely hindered by the fact that standard BIE formulations require the knowledge of problem-specific Green functions to deal with the unboundedness of the material interfaces. These Green functions, however, are often unavailable (in terms of tractable mathematical expressions) or are given in terms of computationally expensive Sommerfeld integrals 1 [27,29,30].Recent advances on BIE methods for time-harmonic problems of scattering from unbounded material interfaces have led to the development of highly efficient solvers that completely bypass the use of problem-specific Green functions [10,12,13,24,30,37]. The windowed Green function (WGF) method, in particular, has successfully...

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Section: Nyström Methods and Frequency-domain Problemsmentioning

confidence: 99%

Section: Nyström Methods and Frequency-domain Problemsmentioning

confidence: 99%

Section: Nyström Methods and Frequency-domain Problemsmentioning

confidence: 99%

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Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

2019

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“…Weighted versions of the double layer and hyper singular operators are defined accordingly [8]. In what follows we express RtR operators associated with quasiperiodic Helmholtz problems using the boundary integral operators introduced above.…”

Section: Dd With Stripes Subdomainsmentioning

confidence: 99%

Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media

2020

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We present a sweeping preconditioner for quasi-optimal Domain Decomposition Methods (DDM) applied to Helmholtz transmission problems in periodic layered media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission operators that are approximations of Dirichlet-to-Neumann (DtN) operators. Employing shape perturbation series, we construct approximations of DtN operators corresponding to periodic domains, which we then use as transmission operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators that are the building blocks of DDM are expressed via robust boundary integral equation formulations. We use Nyström discretizations of quasi-periodic boundary integral operators to construct high-order approximations of RtR. Based on the premise that the quasi-optimal transmission operators should act like perfect transparent boundary conditions, we construct an approximate LU factorization of the tridiagonal QO DD matrix associated with periodic layered media, which is then used as a double sweep preconditioner. We present a variety of numerical results that showcase the effectiveness of the sweeping preconditioners applied to QO DD for the iterative solution of Helmholtz transmission problems in periodic layered media.

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“…§3.1 below) plus an HNA component designed to capture the primary, secondary, and all higher-order corner-diffracted fields. This HNA component takes the form of a sum of terms v j 1 e ik 1 r j and v j 2 e ik 2 r j (see equation (23) below), where k 1 and k 2 are the exterior and interior wavenumbers, r j is Euclidean distance from the jth corner of the polygon, and v j 1 and v j 2 are slowly-varying amplitudes, which are represented by piecewise polynomials on appropriately graded meshes. A BEM implementation was not presented in [31]; instead the performance of the proposed approximation space was investigated by projecting a highly accurate fully numerical reference solution onto the approximation space and computing the resulting approximation error.…”

Section: Introductionmentioning

confidence: 99%

A hybrid numerical–asymptotic boundary element method for high frequency scattering by penetrable convex polygons

2018

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We present a novel hybrid numerical-asymptotic boundary element method for high frequency acoustic and electromagnetic scattering by penetrable (dielectric) convex polygons. Our method is based on a standard reformulation of the associated transmission boundary value problem as a direct boundary integral equation for the unknown Cauchy data, but with a nonstandard numerical discretization which efficiently captures the high frequency oscillatory behaviour. The Cauchy data is represented as a sum of the classical geometrical optics approximation, computed by a beam tracing algorithm, plus a contribution due to diffraction, computed by a Galerkin boundary element method using oscillatory basis functions chosen according to the principles of the Geometrical Theory of Diffraction. We demonstrate with a range of numerical experiments that our boundary element method can achieve a fixed accuracy of approximation using only a relatively small, frequency-independent number of degrees of freedom. Moreover, for the scattering scenarios we consider, the inclusion of the diffraction term provides an order of magnitude improvement in accuracy over the geometrical optics approximation alone.

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Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

Cited by 18 publications

References 52 publications

Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media

A hybrid numerical–asymptotic boundary element method for high frequency scattering by penetrable convex polygons

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