Numerical quadrature for computing of singular integrals

In the present work we have studied superconvergence of Hadamard finite-part integral. We have studied the second-order and the third-order quadrature formulae of Newton-Cotes type. We follow works [Sun, Wu, 2005b], [Lü, Wu, 2005] and work [Wu, Yu and Zhang, 2009] and introduce new rule which gives the same convergence rate as rules in [Lü and Wu, 2005] and [Wu, Yu and Zhang, 2009] but in more general cases. In this work, first we mention the main results in superconvergence of Newton-Cotes rules, we mention trapezoidal and Simpson’s rules and then we introduce rule based on cubic approximation, in the second part we describe important error’s estimates and in the last section we present numerical results for one example. Introduction We want to compute the following integral: Ip(a, b; s, f) = ∫ b a f(t) (t− s)p+1 dt, s ∈ (a, b), p = 1, 2. (1) The evaluation of these integrals plays importance role in boundary element methods and in differential equations. Newton-Cotes rules are quite easy for implementation and their main advantage against Gaussian method is that there is less mesh restriction. The global convergence rate of Newton-Cotes methods is lower for Hadamard finite-part integrals than for the Riemann integrals in general, but from works of Sun W. and Wu J. [2005b] and Lü Y. and Wu J. [2005] we know that there exist points, in which the convergence is better than in general. There is more definitions of (1). We use definition from [Kythe and Schaeferkotter, 2005]: Ip(a, b; s, f) = lim ε→0+ { ∫ s−ε a f(t) (t− s)p+1 dt + ∫ b s+ε f(t) (t− s)p+1 dt− 2f (p−1)(s) ε } , where s ∈ (a, b). We know that the limit exists if f (p)(t) is Hölder continuous on [a,b]. We do uniform partition of interval [a, b] getting a = t0 < . . . < tn = b and we define h = (b − a)/n. Now we can define piecewise linear interpolation of f(t) such as: fL(t) = t− ti−1 h f(ti) + ti − t h f(ti−1), t ∈ [ti−1, ti], i = 1, . . . , n, (2) where we can see that it is Lagrange linear interpolation. If we take fL(t) into (1) instead of f(t), we get a composite trapezoidal rule which has a form: ∫ b a fL(t) (t− s)p+1 dt = n ∑ i=0 ω i (s)f(ti), p = 1, 2. (3) We can take non-uniform partition and we get similarly rule but we have taken uniform partition for simplicity in error estimates. We can also use piecewise quadratic interpolation instead of linear. We define: fQ(t) = 2(t− ti−1/2)(t− ti) h2 f(ti−1)− 4(t− ti−1)(t− ti) h2 f(ti−1/2) + + 2(t− ti−1)(t− ti−1/2) h2 f(ti), t ∈ [ti−1, ti], i = 1, . . . , n, (4) 166 WDS'09 Proceedings of Contributed Papers, Part I, 166–170, 2009. ISBN 978-80-7378-101-9 © MATFYZPRESS