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Implementation of a boundary element method for high frequency scattering by convex polygons
 ADVANCES IN BOUNDARY INTEGRAL METHODS (PROCEEDINGS OF THE 5TH UK CONFERENCE ON BOUNDARY INTEGRAL METHODS
"... In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. H ..."
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Cited by 43 (20 self)
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In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretisation of a wellknown second kind combinedlayerpotential integral equation. We provide a proof that this equation and its adjoint are wellposed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.
Condition Number Estimates for Combined Potential Boundary Integral Operators in Acoustic Scattering
"... We study the classical combined field integral equation formulations for timeharmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to BrakhageWerner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower a ..."
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Cited by 37 (4 self)
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We study the classical combined field integral equation formulations for timeharmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to BrakhageWerner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single and doublelayer potential operators.
CONVERGENCE ANALYSIS FOR FINITE ELEMENT DISCRETIZATIONS OF THE HELMHOLTZ EQUATION WITH DIRICHLETTONEUMANN BOUNDARY CONDITIONS
"... Abstract. A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in R d, d ∈{1, 2, 3} is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasioptimality of the method. As an application of the general theory, ..."
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Cited by 21 (7 self)
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Abstract. A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in R d, d ∈{1, 2, 3} is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasioptimality of the method. As an application of the general theory, a full error analysis of the classical hpversion of the finite element method (hpFEM) is presented for the model problem where the dependence on the mesh width h, the approximation order p, andthewavenumberk is given explicitly. In particular, it is shown that quasioptimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k). 1.
Numerical estimation of coercivity constants for boundary integral operators in acoustic scattering
, 2010
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A New Frequencyuniform Coercive Boundary Integral Equation for Acoustic Scattering
, 2011
"... A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2.� / (where � is the surface of the scatte ..."
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Cited by 14 (2 self)
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A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2.� / (where � is the surface of the scatterer) for all Lipschitz starshaped domains. Moreover, the coercivity is uniform in the wavenumber k D!=c,where! is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “starcombined ” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors’ knowledge, it is the only secondkind integral operator for which convergence of the Galerkin method in L2.� / is proved without smoothness assumptions on � except that it is Lipschitz. The coercivity of the starcombined operator implies frequencyexplicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the highfrequency case. The proof of coercivity of the starcombined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.
Preasymptotic error analysis of CIPFEM and FEM for the Helmholtz equation with high wave number. Part I: linear version
, 2014
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A High Frequency hp Boundary Element Method For SCATTERING BY CONVEX POLYGONS
, 2011
"... In this paper we propose and analyse a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by soundsoft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymp ..."
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Cited by 10 (5 self)
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In this paper we propose and analyse a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by soundsoft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.