Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integ...

We study the construction of a quadrature rule which allows the simultaneous integration of a given function with respect to different weights. This construction is built on the basis of simultaneous Padé approximation of a Nikishin system of functions. The properties of these approximants are used in the proof of convergence of the quadratures and positivity of the corresponding quadrature coe...

In this paper, we discuss a family of multipoint flux mixed finite element (MFMFE) methods on simplicial, quadrilateral, hexahedral, and triangular-prismatic grids. The MFMFE methods are locally conservative with continuous normal fluxes, since they are developed within a variational framework as mixed finite element methods with special approximating spaces and quadrature rules. The latter all...

We address the problem of computing IMSE-optimal designs for random field interpolation models. A spectral representation of the IMSE criterion is obtained from the eigendecomposition of the integral operator defined by the covariance kernel of the random field and integration measure considered. The IMSE can then be approximated by spectral truncation and bounds on the error induced by this tr...

We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function ei!x on the interval [ 1, 1]. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency !. However, accuracy is maintained for all values of ! and in particular the rule elegantly reduces ...

This paper considers the problem of approximation of a continuous function on the unit sphere S ⊆ IR by a spherical polynomial from the space IPn of all spherical polynomials of degree ≤ n. For r = 3 it was shown in [16] that the hyperinterpolation approximation Lnf (which is the linear projection obtained by approximating the Fourier coefficients in the exact L2 orthogonal projection by a posi...

Nowadays, the design of so-called consistent time-stepping schemes that basically feature a physically correct time integration, is still a state-of-the-art topic in the area of numerical mechanics. Within the proposed framework for finite elastoplasto-dynamics, the spatial as well as the time discretisation rely both on a Finite Element approach and the resulting algorithmic conservation prope...

A quadrature-rule type method is presented to approximate the quasicontinuum method for atomistic mechanics. For both the short-range and long-range interaction cases, the complexity of this method depends on the number of representative particles but not on the total number of particles. Simple analysis and numerical experiments are provided to illustrate the accuracy and performance of the me...

In this paper, we propose a natural collocation method with exact imposition of mixed boundary conditions based on a generalized Gauss-Lobatto-Legendre-Birhoff quadrature rule that builds in the underlying boundary data. We provide a direct construction of the quadrature rule, and show that the collocation method can be implemented as efficiently as the usual collocation scheme for PDEs with Di...

In the last ten years, the traditional Simpson quadrature rule for numerical integration has been improved to an optimal 3-point quadrature formula and a modified Simpson rule that takes additionally the first derivative of the approximated integrand at the two end-points of integration into account. The impact of the improved quadrature rules on the fast Fourier transform (FFT) approximation o...