A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov–Galerkinmethod with quadrature, for solving second-order, variable coefficient,elliptic boundary value problems on rectangular domains. Inour scheme, the trial space consists of C² splines of degreer 3, the test space consists of C⁰ splines of degree r –2, and we use composite (r – 1)-point Gauss quadrature.We show existence and uniqueness of the approximate solutionand establish optimal order error bounds in H², H¹ and L² norms.     

A Unified Analysis of Quaternary DS-CDMA Systems with Arbitrary Chip Waveforms and Its Applications

This paper is to present a systematic performance analysis of asynchronous quaternary direct sequence code division multiple access (DS-CDMA) systems using random signature sequences with arbitrary chip waveforms. The simplified improved Gaussian approximation method for bit error rate computation is extended to include arbitrary time-limited (full response or partial response) or band-limited chip waveforms with arbitrary receiver filters. As a time-limited partial response chip waveform modulation format, the well-known power and spectral efficient superposed quadrature amplitude modulation with matched filter or zero-forcing filter is evaluated, and the results show that the optimum zero-forcing filter will yield a performance better than the matched filter counterpart. For band-limited chip waveforms, based on an elementary density function of a second-order polynomial, a class of second-order continuity pulses is proposed for analysis. It is found that all common band-limited pulses are only its special cases. As a member of the class, the widely used frequency domain raised cosine pulse has the worst anti-multiuser-access-interference capability, which has been pointed out in (H. H. Nguyen, Proceedings of IEEE Canadian Conference on Electrical & Computer Engineering, 2002, pp. 1271–1275).     

The best quadrature formula based on Hermite information for the class <Emphasis Type="Italic">KW</Emphasis><Superscript>2</Superscript> [<Emphasis Type="Italic">a,b</Emphasis>]

The best quadrature formula has been found in the following sense: for a function whose norm of the second derivative is bounded by a given constant and the best quadrature formula for the approximate evaluation of integration of that function can minimize the worst possible error if the values of the function and its derivative at certain nodes are known. The best interpolation formula used to get the quadrature formula above is also found. Moreover, we compare the best quadrature formula with the open compound corrected trapezoidal formula by theoretical analysis and stochastic experiments.     

Tight Gaussian 4-Designs

A Gaussian t-design is defined as a finite set X in the Euclidean space ℝⁿ satisfying the condition: for any polynomial f(x) in n variables of degree at most t, here α is a constant real number and ω is a positive weight function on X. It is easy to see that if X is a Gaussian 2e-design in ℝⁿ, then . We call X a tight Gaussian 2e-design in ℝⁿ if holds. In this paper we study tight Gaussian 2e-designs in ℝⁿ. In particular, we classify tight Gaussian 4-designs in ℝⁿ with constant weight or with weight . Moreover we classify tight Gaussian 4-designs in ℝⁿ on 2 concentric spheres (with arbitrary weight functions).     

On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators

In this paper we set out to understand Filon-type quadratureof highly-oscillating integrals of the form ₀¹ f(x) e^ig(x) dx,where g is a real-valued function and >> 1. Employingad hoc analysis, as well as perturbation theory, we demonstratethat for most functions g of interest the moments behave asymptoticallyaccording to a specific model that allows for an optimal choiceof quadrature nodes. Filon-type methods that employ such quadraturenodes exhibit significantly faster decay of the error for highfrequencies . Perhaps counterintuitively, as long as optimalquadrature nodes are used, rapid oscillation leads to significantlymore precise and more affordable quadrature.     

Tensor product Gauss-Lobatto points are Fekete points for the cube

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the -dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral methods based on Gauss-Lobatto points to non-tensor-product domains, since Fekete points are known to exist and have been computed in the triangle and tetrahedron. In one dimension this result was proved by Fejér in 1932, but the extension to higher dimensions in non-trivial.

     

Multi-Dimensional Quadrature of Singular and Discontinuous Functions

In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.     

Analysis of a class of nonconforming finite elements for crystalline microstructures

An analysis is given for a class of nonconforming Lagrange-type finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure. These elements were first proposed and analyzed in 1992 by Rannacher and Turek for the Stokes equation. Our analysis highlights the features of these elements which make them effective for the computation of microstructure. New results for superconvergence and numerical quadrature are also given.

     

Numerical solution of modified Black–Scholes equation pricing stock options with discrete dividend

This paper deals with the numerical solution of the modified Black–Scholes equation modelling the valuation of stock options with discrete dividend payments. By using a delta-defining sequence of the involved generalized Dirac delta function and applying the Mellin transform, an integral formula for the solution is obtained. Then, numerical quadrature approximations and illustrative examples are given.     

Gaussian quadrature and acceleration of convergence

The aim of this paper is to take up again the study done in previous papers, to the case where the integrand possesses an algebraic singularity within the interval of integration. The singularities or poles close to the interval of integration considered in this paper are only real or purely imaginary. This revised version was published online in August 2006 with corrections to the Cover Date.