2,191 research outputs found
A higher-order singularity subtraction technique for the discretization of singular integral operators on curved surfaces
This note is about promoting singularity subtraction as a helpful tool in the
discretization of singular integral operators on curved surfaces. Singular and
nearly singular kernels are expanded in series whose terms are integrated on
parametrically rectangular regions using high-order product integration,
thereby reducing the need for spatial adaptivity and precomputed weights. A
simple scheme is presented and an application to the interior Dirichlet Laplace
problem on some tori gives around ten digit accurate results using only two
expansion terms and a modest programming- and computational effort.Comment: 7 pages, 2 figure
Planewave density interpolation methods for 3D Helmholtz boundary integral equations
This paper introduces planewave density interpolation methods for the
regularization of weakly singular, strongly singular, hypersingular and nearly
singular integral kernels present in 3D Helmholtz surface layer potentials and
associated integral operators. Relying on Green's third identity and pointwise
interpolation of density functions in the form of planewaves, these methods
allow layer potentials and integral operators to be expressed in terms of
integrand functions that remain smooth (at least bounded) regardless the
location of the target point relative to the surface sources. Common
challenging integrals that arise in both Nystr\"om and boundary element
discretization of boundary integral equation, can then be numerically evaluated
by standard quadrature rules that are irrespective of the kernel singularity.
Closed-form and purely numerical planewave density interpolation procedures are
presented in this paper, which are used in conjunction with Chebyshev-based
Nystr\"om and Galerkin boundary element methods. A variety of numerical
examples---including problems of acoustic scattering involving multiple
touching and even intersecting obstacles, demonstrate the capabilities of the
proposed technique
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