Results 1  10
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37
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.
Geometrical structure of Laplacian eigenfunctions
, 2013
"... We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and com ..."
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Cited by 11 (3 self)
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We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is placed onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.
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.
Coercivity of Combined Boundary Integral Equations in HighFrequency Scattering
, 2014
"... We prove that the standard secondkind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e. signdefinite) for all smooth convex domains when the wavenumber k is sufficient large. (This integral equation involves the socalled “combined potent ..."
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Cited by 8 (3 self)
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We prove that the standard secondkind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e. signdefinite) for all smooth convex domains when the wavenumber k is sufficient large. (This integral equation involves the socalled “combined potential ” or “combined field ” operator.) This coercivity result yields kexplicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numericalasymptotic methods developed recently). Coercivity also gives kexplicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewisepolynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation.
High frequency scattering by convex curvilinear polygons
, 2008
"... We consider scattering of a timeharmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with Lipschitz boundary. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the number of degrees of freedom required to achieve a prescrib ..."
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Cited by 6 (4 self)
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We consider scattering of a timeharmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with Lipschitz boundary. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave. Here we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes; a uniform mesh on illuminated sides, and graded meshes refined towards the corners of the polygon on illuminated and shadow sides. Numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases.
An explicit kernelsplit panelbased Nyström scheme for integral equations on axially symmetric surfaces
 J. Comput. Phys
, 2014
"... boundary value problems ..."