Discretization by finite elements of a model parameter dependent problem

The discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy. It is shown that the approximation achieved by a standard finite element method degenerates for thin beams. In contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter. The most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by Gauss quadratures.     

Strong uniqueness and second order convergence in nonlinear discrete approximation

Summary Strong uniqueness has proved to be an important condition in demonstrating the second order convergence of the generalised Gauss-Newton method for discrete nonlinear approximation problems [4]. Here we compare strong uniqueness with the multiplier condition which has also been used for this purpose. We describe strong uniqueness in terms of the local geometry of the unit ball and properties of the problem functions at the minimum point. When the norm is polyhedral we are able to give necessary and sufficient conditions for the second order convergence of the generalised Gauss-Newton algorithm.     

Comparison theorems for compound quadrature formulas

Summary The aim of this note is to compare the rates of convergence of quadrature processes. To this end, representation formulas for the remainders of one process in terms of a second one are developed. The main tool is the Möbius inversion. It turns out that for a large class of compound quadrature processes the rates of convergence are essentially the same.     

An inverse problem in population dynamics

The evolution of population densities of two interacting species in presence of diffusion phenomena is governed by a system of semilinear Volterra integrodifferential parabolic equations. In this system there are time convolution integrals, accounting for past history effects, which are essentially characterized by kernels depending on time only. These delay kernels can be viewed as entries of a 2x2 matrix K. The inverse problem of determining K via suitable population measurements is analyzed.     

Approximation of general arch problems by straight beam elements

Summary In this paper, we approximate the solution of a problem of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration. Compatibility conditions which have to be satisfied at the mesh points are given. These conditions ensure for this method the same order of convergence as usual conforming finite element methods.     

On Gauss quadrature formulas with equal weights

Summary It is well known that the Chebyshev weight function (1–x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe all weight functions for which thenm point Gauss quadrature formula has equal weights for alln, wherem is fixed.     

On integrating vertex singularities using extrapolation

A new approach to the integration of vertex singularities is described. This approach is based on a non-uniform subdivision of the region of integration and the technique fits well to the subdivision strategy used in many adaptive algorithms. A nice feature with this approach is that it can be used in any dimension and on any region of integration which can be subdivided into subregions of the same form. The strategy can be applied both to vertex singularities and internal point singularities. In the latter case this can be done without an initial subdivision of the region in order to put the singular point in a vertex. It turns out that the technique has excellent numerical stability properties.Dedicated to Carl-Erik Fröberg on the occasion of his 75th birthday.This work was supported by The Norwegian Research Council for Science and the Humanities.     

Discrete cubic spline interpolation

Summary In the present paper we study the existence, uniqueness and convergence of discrete cubic spline which interpolate to a given function at one interior point of each mesh interval. Our result in particular, includes the interpolation problems concerning continuous periodic cubic splines and discrete cubic splines with boundary conditions considered respectively in Meir and Sharma (1968) and Lyche (1976) for the case of equidistant knots.     

Some results about the pseudospectral approximation of one-dimensional fourth-order problems

Summary We analyze the pseudospectral approximation of fourth order problems. We give convergence results in the one dimensional case. Numerical experiments are shown in two dimensions for the approximation of the rhombic plate bending problem. Eigenvalues and preconditioning are also investigated.Work performed in the research program of the Istituto di Analisi Numerica del C.N.R., Pavia (Italy)The second author has been sponsored by Deutsche Forschungsgemeinschaft, contract He 1624/1-1     

Aitken's and Steffensen's accelerations in several variables

Summary Aitken's acceleration of scalar sequences extends to sequences of vectors that behave asymptotically as iterations of a linear transformation. However, the minimal and characteristic polynomials of that transformation must coincide (but the initial sequence of vectors need not converge) for a numerically stable convergence of Aitken's acceleration to occur. Similar results hold for Steffensen's acceleration of the iterations of a function of several variables. First, the iterated function need not be a contracting map in any neighbourhood of its fixed point. Instead, the second partial derivatives need only remain bounded in such a neighbourhood for Steffensen's acceleration to converge quadratically, even if ordinary iterations diverge. Second, at the fixed point the minimal and characteristic polynomials of the Jacobian matrix must coincide to ensure a numerically stable convergence. By generalizing the work that Noda did on the subject between 1981 and 1986, the results presented here explain the numerical observations reported by Henrici in 1964 and 1982.This work was supported by a grant from the Northwest Institute for Advanced Study, an organ of Eastern Washington University     

Comparison of optimal quadrature formulas

Summary We prove the monotonicity of the error of the optimal quadrature formula of a given quasi-Hermitian type inW _q ^r [0.1] (1<q) with respect to the order of the derivatives appearing in the end point terms.     

Analysis of least squares finite element methods for a parameter-dependent first-order system *

We consider polynomials orthagonal with respect to a measure μ with an absolutely continuous component and a finite discrete part. We prove that subject to certatin integrability conditions, the polynomials satisfy a second order differential equation. The zeroes of such polynomials determine the equilibrium position of movable n unit charges in an external field determined by the measure μ. We also evaluate the discriminant of such orthagonal polynomials and use it to compute the total energy of the system at equilibrium in terms of the recursion coefficients of the orthonormal polynomials. We also investigate several explicit models, the Koornwinder polynomials, the Ginzburg-Landau potential and the generalized Jacobi weights.     

Complete spline smoothing

Summary The purpose of this paper is to develop complete spline smoothing methods from a computational point of view, culminating in efficient stable numerical algorithms. Both the univariate and bivariate (tensorproduct) cases will be treated.Supported in part by NASA Contract NAS9-16664     

On theoretical and numerical aspects of the bang-bang-principle

Summary In the first part of this paper we are dealing with theoretical statements and conditions which finally lead to bang-bang-principles. A careful analysis of these theorems is used for the development of a numerical method. This method consists of two stages: During the first iterations the number and approximate location of the switching points of the optimal control are determined. In the second phase a rapidly convergent algorithm determines the exact location. We apply this method successfully to a parabolic boundary control problem and give an extensive discussion of numerical results.The work of the second author on this paper was partially done during his stay at North Carolina State University, Graduate Program in Operations Research and Department of Mathematics, Raleigh, USA     

Jensen's Inequality for polynomials with concentration at low degrees

Summary We give a generalization of Jensen's Inequality, valid for polynomials having some concentration at low degrees. We investigate the constants involved, both from a theoretical and a numerical point of view.     

Well-posedness,stability and numerical results for the thermoelastic behavior of a coupled joint-beam PDE-ODE system modeling the transverse motions of the antennas of a space structure

A mathematical model for both axial and transverse motions of two beams with cylindrical cross-sections coupled through a joint is presented and analyzed. The motivation for this problem comes from the need to accurately model damping and joint dynamics for the next generation of inflatable/rigidizable space structures. Thermo-elastic damping is included in the two beams and the motions are coupled through a joint which includes an internal moment. Thermal response in each beam is modeled by two temperature fields. The first field describes the circumferentially averaged temperature along the beam, and is linked to the axial deformation of the beam. The second describes the circumferential variation and is coupled to transverse bending. The resulting equations of motion consist of four, second-order in time, partial differential equations, four, first-order in time, partial differential equations, four second order ordinary differential equations, and certain compatibility boundary conditions. The system is written as an abstract differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections and temperature fields, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove that the system is well-posed, and that with positive damping parameters the resulting semigroup is exponentially stable. Steady states are characterized and several numerical approximation results are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Incomplete data problems in X-ray computerized tomography

Summary In the present paper we study truncated projections for the fanbeam geometry in computerized tomography. First we derive consistency conditions for the divergent beam transform. Then we study a singular value decomposition for the case where only the interior rays in the fan are provided, as for example in region-of-interest tomography. We show that the high angular frequency components of the searched-for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition.The work of the authors was supported by the Deutsche Forschungsgemeinschaft under grant Lo 310/2-4     

An algorithm for the computation of the exponential spline

Summary Procedures for the calculation of the exponential spline (spline under tension) are presented in this paper. The procedureexsplcoeff calculates the second derivatives of the exponential spline. Using the second derivatives the exponential spline can be evaluated in a stable and efficient manner by the procedureexspl. The limiting cases of the exponential spline, the cubic spline and the linear spline are included. A proceduregenerator is proposed, which computes appropriate tension parameters. The performance of the algorithm is discussed for several examples.Editor's Note: In this fascile, prepublication of algorithms from the Approximation series of the Handbook for Automatic Computation is continued. Algorithms are published in ALGOL 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones     

Error analysis for absorbing boundary conditions

Summary An estimate is derived for the error committed by the introduction of artificial boundaries and corresponding artificial boundary conditions when solving wave equations on unbounded domains. The estimate has two terms. One is proportional to the largest reflection coefficient for the artificial boundary condition, the maximum taken only on those rays which appear in the computation. The second term is proportional to 1/k wherek is a measure of the average frequency present in the solution.Research partially supported by the U.S. National Science Foundation under grant number DMS-8601783     

Product integration of singular integrands based on cubic spline interpolation at equally spaced nodes

Summary In this paper the convergence of product integration rules, based on cubic spline interpolation at equally spaced nodes, with not-a-knot end condition, is investigated for integrand functions with a interior or endpoint singularity in the integration interval.Work sponsored by the Ministero della Pubblica Istruzione of Italy