Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod extension involves evaluating the derivative of the integrand, a new extension is introduced using n + 2 points which requires only values of the integrand. However, this new rule does not exist for all n...

It is shown that the Kronrod extension to the «-point Gauss integration rule, with respect to the weight function (1 x2)V~"2, 0 < M < 2, ju i= 1, is of exact precision 3n + 1 for n even and 3n + 2 for n odd. Similarly, for the (n-t-l)-point Lobatto rule, with — V¡ < M < 1, u ^ 0, the exact precision is 3n for n odd and 3n + 1 for n even.

Computational methods are developed for generating Gauss-type quadrature formulae having nodes of arbitrary multiplicity at one or both end points of the interval of integration. Positivity properties of the boundary weights are investigated numerically, and related conjectures are formulated. Applications are made to moment-preserving spline approximation. AMS subject classification: 65D30.

Optimal a priori error bounds are theoretically derived, and numerically verified, for approximate solutions to the 2D homogeneous wave equation obtained by spectral element method. To be precise, method studied here takes advantage of Gauss-Lobatto-Legendre quadrature, thus resulting in under-integrated elements but diagonal mass matrix. The approximation H1 is shown order O(hp) with respect s...

Several old and new finite-element preconditioners for nodal-based spectral discretizations of −∆u = f in the domain Ω = (−1, 1) (d = 2 or 3), with Dirichlet or Neumann boundary conditions, are considered and compared in terms of both condition number and computational efficiency. The computational domain covers the case of classical single-domain spectral approximations (see [5]), as well as t...

In this paper, we consider the second-kind Chebyshev polynomials (SKCPs) for the numerical solution of the fractional optimal control problems (FOCPs). Firstly, an introduction of the fractional calculus and properties of the shifted SKCPs are given and then operational matrix of fractional integration is introduced. Next, these properties are used together with the Legendre-Gauss quadrature fo...

We complete and extend the asymptotic analysis of the spectrum of Jacobi Tau approximations that were first considered by Dubiner. The asymptotic formulas for Jacobi polynomials N P ( , ) , , 1 > − α β α β are derived and confirmed by numerical approximations. More accurate results for the slowest decaying mode are obtained. We explain where the large negative eigenvalues come from. Furthermore...

Solutions of partial differential equations with coordinate singularities often have special behavior near the singularities, which forces them to be smooth. Special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency. It has been observed numerically in the past that, for a scheme to attain high accuracy,...