This paper deals with an extension of one-dimensional Clenshaw–Curtis quadrature rule to R ; d P 2 on a convex domain. As one of its applications, we apply this quadrature rule to a collocation least-squares method using arbitrary abscissas for a first-order system of an elliptic boundary value problem so that the convergence analysis on a numerical solution can be shown in a standard Sobolev n...

We consider multivariate integration in the weighted spaces of functions with mixed first derivatives bounded in Lp norms and the weighted coefficients introduced via `q norms, where p, q ∈ [1,∞]. The integration domain may be bounded or unbounded. The worst-case error and randomized error are investigated for quasi-Monte Carlo quadrature rules. For the worst-case setting the quadrature rule us...

We study a semi-discrete Galerkin method for solving the single-layer equation Vu = f with an approximating subspace of piecewise constant functions. Error bounds in Sobolev norms kkk s with ?1 s < 1 2 are proven and are of the same order as for the original Galerkin method. The distinctive features of the present work are that we handle irregular meshes and do not rely on Fourier methods. The ...

A Sine function approach is used to derive a new Hunter type quadrature rule for the evaluation of Cauchy principal value integrals of certain analytic functions. Integration over a general arc in the complex plane is considered. Special treatment is given to integrals over the interval (-1, 1). It is shown that the quadrature error is of order 0(e~ ), where N is the number of nodes used, and w...

Cranley and Patterson put forward the following randomization as the basis for the estimation of the error of a lattice rule for an integral of a one-periodic function over the unit cube in s dimensions. The lattice rule is randomized using independent random shifts in each coordinate direction that are uniformly distributed in the interval [0, 1]. This randomized lattice rule results in an unb...

Let d and k be positive integers. Let μ be a positive Borel measure on R2 possessing moments up to degree 2d − 1. If the support of μ is contained in an algebraic curve of degree k, then we show that there exists a quadrature rule for μ with at most dk many nodes all placed on the curve (and positive weights) that is exact on all polynomials of degree at most 2d − 1. This generalizes both Gauss...