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29
On Nonreflecting Boundary Conditions
 J. COMPUT. PHYS
, 1995
"... Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated ..."
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Cited by 219 (4 self)
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Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated condition. Second, the exact DtN boundary condition is derived for elliptic and spheroidal coordinates. Third, approximate local boundary conditions are derived for these coordinates. Fourth, the truncated DtN condition in elliptic and spheroidal coordinates is modified to remove difficulties. Fifth, a sequence of new and more accurate local boundary conditions is derived for polar coordinates in two dimensions. Numerical results are presented to demonstrate the usefulness of these improvements.
Numerical Solution Of Problems On Unbounded Domains. A Review
 A review, Appl. Numer. Math
, 1998
"... While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears most significant in many ..."
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Cited by 126 (19 self)
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While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems represent a wide class of important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms and interpretation of the results. Most of the currently used techniques for setting the ABC's can basically be classified into two groups. The methods from the first group (global ABC's) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computa...
Multiscale and stabilized methods
 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
, 2004
"... This article presents an introduction to multiscale and stabilized methods, which represent unied approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have be ..."
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Cited by 23 (9 self)
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This article presents an introduction to multiscale and stabilized methods, which represent unied approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of finite difference, nite volume, and spectral methods, in addition to nite element methods.) The analytical ideas are rst illustrated for timeharmonic wavepropagation problems in unbounded fluid domains governed by the Helmholtz equation. This leads to the wellknown DirichlettoNeumann formulation. A general treatment of the variational multiscale method in the context of an abstract Dirichlet problem is then presented which is applicable to advectivediffusive processes and other processes of physical interest. It is shown how the exact theory represents a paradigm for subgridscale models and a posteriori error estimation. Hierarchical pmethods and bubble function methods are examined in order to understand and, ultimately, approximate the "finescale Green's function " which appears in the theory. Relationships among socalled residualfree bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of nonsymmetric, linear evolution operators formulated in space
hpDISCONTINUOUS GALERKIN METHODS FOR THE HELMHOLTZ EQUATION WITH LARGE WAVE NUMBER
, 2009
"... This paper develops some interior penalty hpdiscontinuous Galerkin (hpDG) methods for the Helmholtz equation in two and three dimensions. The proposed hpDG methods are defined using a sesquilinear form which is not only meshdependent but also degreedependent. In addition, the sesquilinear form ..."
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Cited by 20 (3 self)
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This paper develops some interior penalty hpdiscontinuous Galerkin (hpDG) methods for the Helmholtz equation in two and three dimensions. The proposed hpDG methods are defined using a sesquilinear form which is not only meshdependent but also degreedependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order p. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts. It is proved that the proposed hpdiscontinuous Galerkin methods are absolutely stable (hence, wellposed). For each fixed wave number k, suboptimal order error estimates in the broken H 1norm and the L 2norm are derived without any mesh constraint. The error estimates and the stability estimates are improved to optimal order under the mesh condition k 3 h 2 p −1 ≤ C0 by utilizing these stability and error estimates and using a stabilityerror iterative procedure To overcome the difficulty caused by strong indefiniteness of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [19, 20, 33], which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size h, the polynomial degree p, the wave number k, as well as all the penalty parameters for the numerical solutions.
A stable, highorder method for twodimensional boundedobstacle scattering
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2006
"... A stable and highorder method for solving the Helmholtz equation on a twodimensional domain exterior to a bounded obstacle is developed in this paper. The method is based on a boundary perturbation technique ("transformed field expansions") coupled with a wellconditioned highorder spectr ..."
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Cited by 19 (12 self)
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A stable and highorder method for solving the Helmholtz equation on a twodimensional domain exterior to a bounded obstacle is developed in this paper. The method is based on a boundary perturbation technique ("transformed field expansions") coupled with a wellconditioned highorder spectralGalerkin solver. The method is further enhanced with numerical analytic continuation, implemented via Padé approximation. Ample numerical results are presented to show the accuracy, stability, and versatility of the proposed method.
Analysis of a spectralGalerkin approximation to the Helmholtz equation in exterior domains
 SIAM J. Numer. Anal
"... Abstract. An error analysis is presented for the spectralGalerkin method to the Helmholtz equation in 2 and 3dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the DirichlettoNeumann operator, and then a spectralGalerkin method ..."
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Cited by 16 (6 self)
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Abstract. An error analysis is presented for the spectralGalerkin method to the Helmholtz equation in 2 and 3dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the DirichlettoNeumann operator, and then a spectralGalerkin method is employed to approximate the reduced problem. The error analysis is based on exploring delicate asymptotic behaviors of the Hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems. Explicit error bounds with respect to the wave number are derived, and some illustrative numerical examples are also presented. Key words. Helmholtz equation, wave scattering, error analysis, spectralGalerkin, unbounded domain
Recent Developments in Finite Element Methods for Structural Acoustics
, 1996
"... This paper reviews recent progress in finite element analysis that renders computation a practical tool for solving problems of structural acoustics ..."
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Cited by 14 (3 self)
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This paper reviews recent progress in finite element analysis that renders computation a practical tool for solving problems of structural acoustics
A weakly singular form of the hypersingular boundary integral equation applied to 3D acoustic wave problems
, 1992
"... ..."
Fast Numerical Solution Of Exterior Helmholtz Problems With Radiation Boundary Condition By Imbedding
, 1994
"... The development of efficient solution algorithms for Poisson's equation on domains allowing for separation of variables prompted research towards extending these algorithms to domains of general shape. The resulting numerical techniques are known as imbedding methods or capacitance matrix metho ..."
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Cited by 8 (2 self)
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The development of efficient solution algorithms for Poisson's equation on domains allowing for separation of variables prompted research towards extending these algorithms to domains of general shape. The resulting numerical techniques are known as imbedding methods or capacitance matrix methods. In this dissertation, we develop capacitance matrix methods for exterior boundary value problems for the Helmholtz equation, a secondorder elliptic PDE which governs timeharmonic wave propagation. Solutions of exterior Helmholtz problems must satisfy an asymptotic boundary condition at infinity in order to be uniquely determined. We incorporate this boundary condition into the discretization by posing the DirichlettoNeumann (DtN) condition, an exact nonlocal boundary condition, on a circular artificial boundary. A fast Helmholtz solver of complexity O(nm log n) can then be obtained for the resulting discrete problem on an m \Theta n grid when the underlying computational domain is an an...