A fast numerical method for a natural boundary integral equation for the Helmholtz equation

Li, Songhua; Sun, Mingbao

doi:10.1016/j.cam.2008.12.002

Cited by 7 publications

(9 citation statements)

References 6 publications

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“…Theorem 3.1 There exists a positive integer J such that for all j > J, the Galerkin scheme (17) or (19) has a unique solution φ j (θ ) ∈ V j , and there is a constant M > 0 such that…”

Section: Multiscale Galerkin Methodsmentioning

confidence: 99%

“…In [14,17], they obtain wavenumber independent boundary element methods in typical domain, that is, a half-plane and exterior circular domain. Colton and Kress [8] have investigated the numerical solution of the integral equation (4) by Nyström method if the Dirichlet boundary value condition f (x) and the boundary curve ∂ is analytic.…”

Section: Introductionmentioning

confidence: 99%

“…Standard schemes for solving Equation (1) become prohibitively expensive as the wavenumber k increases especially. The development of more efficient numerical schemes for high frequency scattering problems has attracted much recent attention in lots of literature [1,10,14,17]. The use of multiscale methods for integral equations is a subject recently studied by many authors.…”

Section: Introductionmentioning

confidence: 99%

“…The numerical solution of the Helmholtz equation with the trigonometric wavelets are investigated in [17,18]. In this article, using the trigonometric wavelets in [17] as trial functions to discretize the boundary integral equation (4), we discover a matrix compression strategy for multiscale Galerkin methods when the Dirichlet boundary value condition f (x) ∈ C 1 and the boundary curve ∂ is of class C 3 .…”

Section: Introductionmentioning

confidence: 99%

“…In this article, using the trigonometric wavelets in [17] as trial functions to discretize the boundary integral equation (4), we discover a matrix compression strategy for multiscale Galerkin methods when the Dirichlet boundary value condition f (x) ∈ C 1 and the boundary curve ∂ is of class C 3 . The compression strategy has the following advantages: (i) the truncated treatment is simple, (ii) the computational complexity of the truncated matrix is concrete, (ii) the condition number of the truncated matrix is bounded.…”

Section: Introductionmentioning

confidence: 99%

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A compression technique for the boundary integral equation reduced from Helmholtz equation

Lin

Xian³

2013

International Journal of Computer Mathematics

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Applying the trigonometric wavelets and the multiscale Galerkin method, we investigate the numerical solution of the boundary integral equation reduced from the exterior Dirichlet problem of Helmholtz equation by the potential theory. Consequently, we obtain a matrix compression strategy, which leads us to a fast algorithm. Our truncated treatment is simple, the computational complexity and the condition number of the truncated coefficient matrix are bounded by a constant. Furthermore, the entries of the stiffness matrix can be evaluated from the Fourier coefficients of the kernel of the boundary integral equation. Examples given for demonstrating our numerical method shorten the runtime obviously.

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“…Theorem 3.1 There exists a positive integer J such that for all j > J, the Galerkin scheme (17) or (19) has a unique solution φ j (θ ) ∈ V j , and there is a constant M > 0 such that…”

Section: Multiscale Galerkin Methodsmentioning

confidence: 99%