A modified boundary integral equation for solving the exterior Robin problem for the Helmholtz equation in three dimensions

Kleefeld, Andreas

doi:10.1016/j.amc.2012.08.055

Cited by 6 publications

(7 citation statements)

References 11 publications

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“…of almost four using quadratic basis functions, thus leading to superconvergence. Note that the numerical program developed by the third author (see [39]) is an extension of BIEPACK (boundary integral equation package for the solution of integral equations of the second kind arising from the Laplace equation) developed by Atkinson (see [5]) which has been recently used in a variety of problems dealing with the Helmholtz equation (see [35,40,41,42,43,44] among others).…”

Section: Numerical Resultsmentioning

confidence: 99%

The factorization method for the acoustic transmission problem

Anagnostopoulos¹,

Charalambopoulos²,

Kleefeld³

2013

Inverse Problems

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In the present work, the shape reconstruction problem of acoustically penetrable bodies from far-field data corresponding to time-harmonic plane wave incidence is investigated within the framework of the factorization method. Although the latter technique has received considerable attention in inverse scattering problems dealing with impenetrable scatterers, it has not been elaborated for inverse transmission problems with the only exception being a work by the first two authors and co-workers. Aimed at bridging this gap in the field of acoustic scattering, the paper at hand focuses on establishing rigorously the necessary theoretical framework for the application of the factorization method to the inverse acoustic transmission problem. The main outcome of the undertaken investigation is the derivation of an explicit formula for the scatterer's characteristic function, which depends solely on the far-field data feeding the inverse scattering scheme. Extended numerical examples in three dimensions are also presented, where a variety of different surfaces are successfully reconstructed by the factorization method, thus complementing the method's validation from the computational point of view.

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Section: Numerical Resultsmentioning

confidence: 99%

The factorization method for the acoustic transmission problem

Anagnostopoulos¹,

Charalambopoulos²,

Kleefeld³

2013

Inverse Problems

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“…In this section, we explain how to discretize (6) using quadratic interpolation of the boundary, but using piecewise quadratic interpolation with α = (1 − 3/5)/2 (see [22] for the 3D case) instead of quadratic interpolation for the unknown u on each of the n f boundary elements which ultimately leads to the non-linear eigenvalue…”

Section: Discretizationmentioning

confidence: 99%

“…Standard convergence results are available for boundary integral equations of the second kind using boundary element collocation method under suitable assumptions on the boundary (for example the boundary is at least of class C 2 ) and the boundary condition for the Laplace equation (see [5]). Quadratic approximations of the boundary and the boundary function yield cubic convergence (refer to [5] for the Laplace equation and [22] for the Helmholtz equation). In fact, the convergence results can be improved as shown in [22] for the threedimensional case.…”

Section: Superconvergencementioning

confidence: 99%

“…Quadratic approximations of the boundary and the boundary function yield cubic convergence (refer to [5] for the Laplace equation and [22] for the Helmholtz equation). In fact, the convergence results can be improved as shown in [22] for the threedimensional case. However, the exact theoretical convergence rate for the eigenvalue is not known.…”

Section: Superconvergencementioning

confidence: 99%

“…Finally, note that a specific choice of 0 < α < 1/2 can improve the overall convergence rate for smooth boundaries (see [22]), but since we are happy with the cubic convergence rate with the pick α = (1 − 3/5)/2 (a Gauss-quadrature point within the interval [0, 1]), we have not investigated this any further.…”

Section: Superconvergencementioning

confidence: 99%

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The hot spots conjecture can be false: Some numerical examples

Kleefeld¹

2021

Preprint

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The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10 −3 . Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.

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A modified boundary integral equation for solving the exterior Robin problem for the Helmholtz equation in three dimensions

Cited by 6 publications

References 11 publications

The factorization method for the acoustic transmission problem

The factorization method for the acoustic transmission problem

The hot spots conjecture can be false: Some numerical examples

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