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A fast algorithm for spherical grid rotations and its application to singular quadrature. (English) Zbl 1285.65089

Summary: We present a fast and accurate algorithm for evaluating singular integral operators on smooth surfaces that are globally parametrized by spherical coordinates. Problems of this type arise, for example, in simulating Stokes flows with particulate suspensions and in multiparticle scattering calculations. For smooth surfaces, spherical harmonic expansions are commonly used for geometry representation and the evaluation of the singular integrals is carried out with a spectrally accurate quadrature rule on a set of rotated spherical grids. We propose a new algorithm that interpolates function values on the rotated spherical grids via hybrid nonuniform fast Fourier transforms. The algorithm has a small complexity constant, and the cost of applying the quadrature rule is nearly optimal \(\mathcal{O}(p^4\log p)\) for a spherical harmonic expansion of degree \(p\).

MSC:

65R20 Numerical methods for integral equations
45P05 Integral operators
33C55 Spherical harmonics
65T50 Numerical methods for discrete and fast Fourier transforms
65T40 Numerical methods for trigonometric approximation and interpolation

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