On Nonreflecting Boundary Conditions

Grote, Marcus J.; Keller, Joseph B.

doi:10.1006/jcph.1995.1210

Cited by 234 publications

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“…An important issue in the numerical implementation of (9) is the well-posedness of the resulting discrete problem. In [29] two of the authors took up this question and showed rigorously that if the DtN map, T , is suitably modified (see [9,15,17]) then the discrete system is indeed well-posed. From a practical standpoint, however, we did not notice any instabilities in our numerical simulations using the unmodified DtN map and thus we advocate its use in generic computations.…”

Section: An Hp-finite Element Methodsmentioning

confidence: 99%

“…FE recursions are used to compute DtN map terms T n (c.f. Table 19) An important point to make regarding these simulations is that since an ellipse is a "separable" geometry one could implement a DtN map of the form (10) based upon the appropriate eigenfunctions for this geometry (in two dimensions the Mathieu functions [15]). Our new approach presents a significant improvement upon this idea since no additional coding is required: An ellipse is easily expressed as an analytic perturbation of a circle for which our DtN map, T (ε), will converge extremely rapidly.…”

Section: θ )mentioning

confidence: 99%

“…In [28] it was demonstrated that this method can be implemented and coupled to a piecewise linear FEM while not destroying its inherent convergence rate. In [29] it was shown that the resulting discrete problem is well-posed provided that the DtN map is suitably modified (see [9,15,17]). Moreover, this proof did not rely upon the piecewise linear nature of the underlying basis/test functions and thus can be applied equally well to other FEM.…”

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Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements

et al. 2009

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For exterior scattering problems one of the chief difficulties arises from the unbounded nature of the problem domain. Inhomogeneous obstacles may require a volumetric discretization, such as the Finite Element Method (FEM), and for this approach to be feasible the exterior domain must be truncated and an appropriate condition enforced at the far, artificial, boundary. An exact, non-reflecting boundary condition can be stated using the classical DtN-FE method if the Artificial Boundary's shape is quite specific: circular or elliptical. Recently, this approach has been generalized to permit quite general Artificial Boundaries which are shaped as perturbations of a circle resulting in the "Enhanced DtN-FE" method. In this paper we extend this method to a two-dimensional FEM featuring high-order polynomials in order to realize a high rate of convergence. This is more involved than simply specifying high-order test and trial functions as now the scatterer shape and Artificial Boundary must be faithfully represented. This entails boundary elements which conform (to high order) to the true boundary shapes. As we show, this can be accomplished and we realize an arbitrary order FEM without spurious reflections.

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Section: An Hp-finite Element Methodsmentioning

confidence: 99%

Section: θ )mentioning

confidence: 99%

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Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements

et al. 2009

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“…There is abundant literature on different numerical techniques that have been developed for this problem, such as boundary element methods [5], infinite element methods [11], methods using nonreflecting boundary conditions [14], perfectly matched layers (PML) [2], among others. In many of these approaches, an essential step is to solve the following problem:…”

Section: Introductionmentioning

confidence: 99%

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

Shen¹,

Wang

2005

SIAM J. Numer. Anal.

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Abstract.A complete error analysis is performed for the spectral-Galerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Green's functions and does not require any mesh condition to be satisfied. Furthermore, new error estimates are also established for multidimensional radial and spherical symmetric domains. Illustrative numerical results in agreement with the theoretical analysis are presented.

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“…Several other nonlocal ABCs' methodologies for unsteady waves have recently been put forward, most notably [7][8][9][10][11][12][13][14]; see also the survey [15]. In comparison, our approach has a number of distinctive characteristics, besides the restricted temporal nonlocality that does not come at a cost of accuracy.…”

Section: Introductionmentioning

confidence: 99%

Artificial Boundary Conditions for the Numerical Simulation of Unsteady Electromagnetic Waves

Tsynkov¹

2003

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A central characteristic feature of an important class of hyperbolic PDEs in odddimension spaces is the presence of lacunae, i.e., sharp aft fronts of the waves, in their solutions. This feature, which is often associated with the Huygens' principle, is employed to construct accurate non-local artificial boundary conditions (ABCs) for the Maxwell equations. The setup includes the propagation of electromagnetic waves from moving sources over unbounded domains. For the purpose of obtaining a finite numerical approximation the domain is truncated, and the ABCs guarantee that the outer boundary be completely transparent for all the outgoing waves. The lacunae-based approach has earlier been used for the scalar wave equation, as well as for acoustics. In the current paper, we emphasize the key distinctions between those previously studied models and the Maxwell equations of electrodynamics, as they manifest themselves in the context of lacunae-based integration. The extent of temporal nonlocality of the proposed ABCs is fixed and limited, and this is not a result of any approximation, it is rather an immediate implication of the existence of lacunae. The ABCs can be applied to any numerical scheme that is used to integrate the Maxwell equations. They do not require any geometric adaptation, and they guarantee that the accuracy of the boundary treatment will not deteriorate even on long time intervals. The paper presents a number of numerical demonstrations that corroborate the theoretical design features of the boundary conditions. Key words: Electromagnetic waves, Maxwell's equations, unsteady propagation, unbounded domains, truncation, finite computational domain, lacunae, non-deteriorating method, long-term numerical integration.Email address: tsynkov@math.ncsu.edu (S. V. Tsynkov). URL: http://www.math.ncsu.edu/∼stsynkov (S. V. Tsynkov).1 The author gratefully acknowledges support by AFOSR, Grant F49620-01-1-0187, and that by NSF, Grant DMS-0107146. Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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On Nonreflecting Boundary Conditions

Cited by 234 publications

References 0 publications

Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements

Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

Artificial Boundary Conditions for the Numerical Simulation of Unsteady Electromagnetic Waves

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