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Generalized quadrature rules of Gaussian type for numerical evaluation of singular integrals. (English) Zbl 1303.41015

Summary: An efficient method for constructing a class of generalized quadrature formulae of Gaussian type on \((- 1, 1)\) for integrands having logarithmic singularities is developed. That kind of singular integrals are very common in the boundary element method. Several special cases for \(n\)-point quadratures, which are exact on both of the spaces \(\mathcal{P}_{2 n - 2 \ell - 1} [- 1, 1]\) (the space of algebraic polynomials of degree at most \(2 n - 2 \ell - 1\)) and \(\mathcal{L}_{2 \ell - 1} [- 1, 1] = \operatorname{span} \{x^k \log | x | \}_{k = 0}^{2 \ell - 1}\) (the logarithmic space), where \(1 \leq \ell \leq n\), are presented. Regarding a direct connection of these \(2 m\)-point quadratures with \(m\)-point quadratures of Gaussian type with respect to the weight function \(t \mapsto t^{- 1 / 2}\) over \((0, 1)\), the method of construction is significantly simplified. Gaussian quadratures on \((0, 1)\) are exact for integrands of the form \(t \mapsto p(t) + q(t) \log t\), where \(p\) and \(q\) are algebraic polynomials of degree at most \(2 m - \ell - 1\) and \(\ell - 1\)   \((1 \leq \ell \leq 2 m)\), respectively. The obtained quadratures can be used in a software implementation of the boundary element method.

MSC:

41A55 Approximate quadratures
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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