Convergence properties of a class of product formulas for weakly singular integral equations

Abstract

Abstract.We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the typewhere the kernel K(x, y) is weakly singular and the function f{x) has singularities only at the endpoints ± 1 . In particular, when K(x , y) = log \x -y\, K(x ,y) = \x -y\", v > -1, and f(x) has algebraic singularities of the form ( 1 ± x)a , a > -1 , we prove that the uniform rate of convergence of the rules is 0{m~ ~ "log m) in the case of the first kernel, and 0(m~ ~ "~ "logm) if v < 0, or 0(m~ ~ " \o%m) if t> > 0, for the second, where m is the number of points in the quadrature rule.

Error estimates in the numerical evaluation of some BEM singular integrals

“…and recall the convergence estimates obtained in [14] (see also [5,Theorem 6]), [16]. This is the situation that most frequently occurs in BEM applications.…”

“…Recalling the convergence estimate derived in [5,Theorem 6], for the above procedure in the case of I 1 (f 1 ) we will have R n = O(n −2q log n), hence R n,n (f ) = O(n −2q log n) since the function f 1 is analytic in the domain of integration.…”

Polynomial Approximations of Functions with Endpoint Singularities and Product Integration Formulas

“…We also remark that in the particular case examined in [2] there were no advantages in writing the function u(x) in the form (1 -x)a(l +x)^v(x) and considering w(x) = (l-x)a(l+x)^ as a weight function, unless v(x) was itself of class Cq[-1, 1 ] ; but this is not the case for the equations we are considering.…”

“…Since the singular terms of such expansions are of the form (1 ± x)*(1+1/) and (l±xy log'(l±x), I < j, k, j, I = 1, 2, ... , in [2,12,14] it has been sufficient to consider the behavior of the quadrature rules when they were applied to those terms.…”

“…These rules integrate exactly the singular components of the kernels (see, for example, [2,10,12]). To derive accurate uniform convergence estimates for these quadratures, we need to take into account the precise behavior of f(x) in the interval of integration.…”