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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 28 (17 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.
On the Numerical Solution of a Hypersingular Integral Equation for Elastic Scattering from a Planar Crack
 J. Comp. & Appl. Maths
"... In this paper we describe a fully discrete quadrature method for the numerical solution of a hypersingular integral equation of the first kind for the scattering of timeharmonic elastic waves by a cavity crack. We establish convergence of the method and prove error estimates in a Holder space setti ..."
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Cited by 14 (3 self)
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In this paper we describe a fully discrete quadrature method for the numerical solution of a hypersingular integral equation of the first kind for the scattering of timeharmonic elastic waves by a cavity crack. We establish convergence of the method and prove error estimates in a Holder space setting. Numerical examples illustrate the convergence results.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
An Optimal Order Collocation Method for First Kind Boundary Integral Equations on Polygons
, 1995
"... This paper discusses the convergence of the collocation method using splines of any order k for first kind integral equations with logarithmic kernels on closed polygonal boundaries in ..."
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Cited by 9 (2 self)
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This paper discusses the convergence of the collocation method using splines of any order k for first kind integral equations with logarithmic kernels on closed polygonal boundaries in
Solvability and spectral properties of integral equations on the real line: I.Weighted spaces of continuous functions
"... ABSTRACT. We consider the solvability of linear integral equations on the real line, in operator form (λ − K)φ = ψ, where λ ∈ C and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on Lp (R), 1 ≤ p ≤ ∞, and on BC(R). We establish co ..."
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Cited by 7 (3 self)
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ABSTRACT. We consider the solvability of linear integral equations on the real line, in operator form (λ − K)φ = ψ, where λ ∈ C and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on Lp (R), 1 ≤ p ≤ ∞, and on BC(R). We establish conditions on families of operators, {Kk: k ∈ W}, whichensurethatifλ̸ = 0andλφ = Kkφ has only the trivial solution in BC(R), for all k ∈ W,then for 1 ≤ p ≤ ∞, (λ − K)φ = ψ has exactly one solution φ ∈ Lp (R) for every k ∈ W and ψ ∈ Lp (R). The results of considerable generality apply in particular to kernels of the form k(s, t) =κ(s − t)z(t) andk(s, t) =˜κ(s − t)˜z(s, t), where κ, ˜κ ∈ L1 (R), z ∈ L ∞ (R), ˜z ∈ BC(R2)and˜κ(s) = O(s−b) as s  → ∞, for some b> 1. As a significant application we consider the problem of acoustic scattering by a soundsoft, unbounded onedimensional rough surface which we reformulate as a second kind boundary integral equation. Combining the general results of earlier sections with a uniqueness result for the boundary value problem, we establish that the integral equation is wellposed as an equation on Lp (R), 1 ≤ p ≤∞, and on weighted spaces of continuous functions. 1. Introduction. We
Biorthogonal Wavelets for the Direct Integral Formulation of the Heat Equation
, 2000
"... We consider the direct integral formulation of the heat equation in a smooth domain of R 2 with Neumann and Dirichlet boundary conditions. The unknown belongs to an anisotropic Sobolev space of positive order for the Neumann problem and of negative order for the Dirichlet one and is approximated b ..."
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Cited by 7 (1 self)
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We consider the direct integral formulation of the heat equation in a smooth domain of R 2 with Neumann and Dirichlet boundary conditions. The unknown belongs to an anisotropic Sobolev space of positive order for the Neumann problem and of negative order for the Dirichlet one and is approximated by the Galerkin method using an appropriate biorthogonal wavelet basis. The use of such a basis allows to compress the stiffness matrix from O(N 2 ) to O(N ), and to obtain a uniformly bounded condition number. Finally, we show that the compressed scheme converges as fast as the Galerkin. 1 Introduction The boundary element method applied to the heat equation on a smooth domain\Omega of R 2 leads to the resolution of a twodimensional problem [8, 16]. Unfortunately, the stiffness matrix is full, and in general illconditioned. Many authors (see [10, 18, 23]) have introduced new bases made of wavelets to overcome these difficulties, but only, to our knowledge, for elliptic problems. There...
Norms of Inverses, Spectra, and Pseudospectra of Large Truncated WienerHopf Operators and Toeplitz Matrices
, 1997
"... . This paper is concerned with WienerHopf integral operators on L p and with Toeplitz operators (or matrices) on l p . The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply ..."
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Cited by 6 (2 self)
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. This paper is concerned with WienerHopf integral operators on L p and with Toeplitz operators (or matrices) on l p . The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply the invertibility of all sufficiently large truncations and the uniform boundedness of the norms of their inverses. Quantitative statements, such as results on the limit of the norms of the inverses, can be proved in the case p = 2 by means of C algebra techniques. In this paper we replace C algebra methods by more direct arguments to determine the limit of the norms of the inverses and thus also of the pseudospectra of large truncations in the case of general p. Contents 1. Introduction 2 2. Structure of Inverses 4 3. Norms of Inverses 7 4. Spectra 15 5. Pseudospectra 17 6. Matrix Case 22 7. Block Toeplitz Matrices 23 8. Pseudospectra of Infinite Toeplitz Matrices 27 References 30 ...
Two Boundary Element Methods for the clamped plate
, 2000
"... In this paper we retail the approximation of the clamped plate problem by means of two boundary element methods. In both cases, the variational formulation is given on product of Sobolev spaces and we avoid the orthogonality to polynomials of degree one. The use of biorthogonal wavelets on these spa ..."
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Cited by 6 (0 self)
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In this paper we retail the approximation of the clamped plate problem by means of two boundary element methods. In both cases, the variational formulation is given on product of Sobolev spaces and we avoid the orthogonality to polynomials of degree one. The use of biorthogonal wavelets on these spaces leads to a wellconditioned stiffness matrix and reduces strongly the complexity thanks to a compression procedure. The solution of the compressed Galerkin scheme converges to the exact solution at the same rate as for the classic Galerkin method.
Efficient Inversion of the Galerkin Matrix of General Second Order Elliptic Operators with Nonsmooth Coefficients
 Math. Comp
, 2005
"... Abstract. This article deals with the efficient (approximate) inversion of finite element stiffness matrices of general secondorder elliptic operators with L ∞coefficients. It will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices (Hmatrices). Furthermore, nu ..."
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Cited by 4 (2 self)
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Abstract. This article deals with the efficient (approximate) inversion of finite element stiffness matrices of general secondorder elliptic operators with L ∞coefficients. It will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices (Hmatrices). Furthermore, numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity. 1.