Accurate quadrature via line extrapolation and rational approximation with application to boundary integral methods for Stokes flow

• In boundary integral methods, special quadrature methods are needed to approximate layer potentials, integrals where the integrand is singular or sharply peaked for evaluation points on or close to the boundaries. In this paper, we study a method based on extrapolation or interpolation along a line, sometimes called the Hedgehog method. In this method, the layer potential is evaluated with a regular quadrature method for evaluation points along a line, and an approximant is constructed and evaluated in an area of interest where the original layer potential is difficult to evaluate due to it being singular or sharply peaked.We analyze the errors in the Hedgehog method with polynomial approximation, and use this to construct optimal distributions of sample points. Furthermore, rational approximation is introduced in the Hedgehog method, and compared with polynomial approximation. It is found that rational approximation can typically achieve a lower error than polynomial approximation, and does not increase the computational cost of the method significantly. Strategies for avoiding and dealing with spurious poles in rational approximation are discussed.We compare extrapolation (no sample point on the boundary) with interpolation (sample point present) in the Hedgehog method, and find that the error in our example is lower in the interpolation case by around one order of magnitude, compared to the extrapolation case.We consider a specific test case, consisting of two rigid rodlike particles in Stokes flow. Parameter selection and error estimation for the Hedgehog method is discussed for this test case. The accuracy and computational cost of the Hedgehog method is examined, and compared with another special quadrature method, namely quadrature by expansion (QBX). We find that the Hedgehog method should be able to compete with QBX in this context, but further investigation is needed for strict tolerances.

Ämnesord

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

Numerical Analysis

Numerisk analys

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