This paper discusses the connection between boundary quadrature formulas constructed by using solutions of partial differential equations and boundary element schemes. AMS Subject Classfication: 65D30, 65D32, 65N38.

Abstract. To find the approximate solutions of a weakly singular integral equation by the collocation method it is necessary to solve linear systems whose coefficients are expressed as integrals. These integrals cannot usually be computed exactly. We get the fully discrete collocation method when we approximate the integrals by quadrature formulas on nonuniform grid. In this paper an appropriat...

In this paper we investigate the Szegő-Radau and Szegő-Lobatto quadrature formulas on the unit circle. These are (n + m)-point formulas for which m nodes are fixed in advance, with m = 1 and m = 2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials. That means that the free parameters (free nodes and positive weights) are chosen such that the quadrature...

It is well known [28] that a number of important classes of univalent functions (e.g. convex, starlike) are related through their derivatives by functions with positive real part. These functions play an important part in problem from signal theory, in moment problems and in constructing quadrature formulas, see Ronning [97] and the references cited therein for some recent applications. In this...

We use a version of the FEM–BEM method introduced by Costabel [Boundary Elements IX, Vol. 1, C. A. Brebbia et al., eds., Springer-Verlag, 1987] and Han [J. Comput. Math., 8 (1990), pp. 223–232] to discretize an exterior quasilinear problem. We provide error estimates for the Galerkin method and propose a fully discrete scheme based on simple quadrature formulas. Furthermore, we show that these ...

In many applications of ageand size-structured population models, there is an interest in obtaining good approximations of total population numbers rather than of their densities. Therefore, it is reasonable in such cases to solve numerically not the PDE model equations themselves, but rather their integral equivalents. For this purpose quadrature formulae are used in place of the integrals. Si...

A number of facts about quadratic algebraic problems and applied Newton–Raphson like methods are presented. The main results are about solution structure, loading trajectories, load flow feasibility boundaries and Newton–Raphson solutions. Various practically important applications of these properties are discussed. Although motivated by power flow problems, the results are valid for any proble...

In this paper we derive accurate numerical methods for the quantum Boltzmann equation for a gas of interacting bosons. The schemes preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and generalized Bose-Einstein distributions as steady states. These properties are essential in order to develop schemes that are able to ca...

Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of π/2 with respect to each space dimension. We propose new nonpolynomial quadrature methods that avoid th...