On a numerical approach to Stefan-like problems

Summary This paper is devoted to the numerical analysis of a bidimensional two-phase Stefan problem. We approximate the enthalpy formulation byC ⁰ piecewise linear finite elements in space combined with a semi-implicit scheme in time. Under some restrictions related to the finite element mesh and to the timestep, we prove positivity, stability and convergence results. Various numerial tests are presented and discussed in order to show the accuracy of our scheme.This work is supported by the Fonds National Suisse pour la Recherche Scientifique.     

Extrapolation to the limit for numerical solutions of hyperbolic equations

Summary The application of extrapolation to the limit requires the existence of an asymptotic expansion in powers of the step size. In this paper one-and multi-step methods for the solution of hyperbolic systems of first order are considered. Conditions are formulated that ensure the asymptotic expansion. Methods of characteristics for quasilinear systems with two independent variables are included in this presentation. If a rectangular grid is used, also non-quasilinear systems are admissible. The main part of this paper deals with initial value problems. But it is shown that in some exceptional cases asymptotic expansions hold for initial-boundary problems, too.This paper is chiefly based on the author's doctoral thesis [7], written under the direction of Professor R. Bulirsch     

Sharp estimates in terms of moduli of smoothness for compound cubature rules

Summary In 1980 Dahmen-DeVore-Scherer introduced a modulus of continuity which turns out to reflect invariance properties of compound cubature rules effectively. Accordingly, sharp error bounds are derived, the existence of relevant counterexamples being a consequence of a quantitative resonance principle, established previously.     

Approximation by boolean sums of positive linear operators III: Estimates for some numerical approximation schemes

In the present note we intröduce and investigate certain sequences of discrete positive linear operators and Boolean sum modifications of them. The mappings considered are obtained by discretizing a class of transformed convolution-type operators using Gaussian quadrature of appropriate order. For our operators and their modifications we prove pointwise Jackson-type theorems involving the first and second order moduli of smoothness, thus providing new and elegant proofs of earlier results by Timan, Telyakowskii, Gopengauz and DeVore. Due to their discrete structure, optimal order of approximation and ease of computation, the operators appear to be useful for numerical approximation. In an intermediate step we solve an old problem in Approximation Theory; its importance was only recently emphasized in a paper of Butzer.     

Error estimates for Gaussian quadrature formulae

Summary We derive both strict and asymtotic error bounds for the Gauss-Jacobi quadrature formula with respect to a general measure. The estimates involve the maximum modulus of the integrand on a contour in the complex plane. The methods are elementary complex analysis.     

Über optimale Quadraturformeln mit freien Knoten zu Hilberträumen periodischer Funktionen

Summary For some special Hilbert-spaces of periodic analytic functions it is known that quadrature formulae of minimal norm with preassigned equidistant nodes are even so-called Wilf-formulae, i.e. they satisfy necessary conditions for minimal norm with respect to their nodes. By simple examples, however, it can be shown that equidistant Wilf-formulae are not necessarily optimal. In this paper the question of optimality of equidistant nodes in quadrature formulae for rather general Hilbert-spaces of periodic analytic functions is answered by giving sufficient conditions which can be interpreted as conditions on the size of the regularity-regions of the functions belonging to the Hilbert-spaces under consideration. Examples prove these conditions to be quite sharp.In addition the trapezoidal-rule is shown to be only optimal formula (with respect to the nodes and coefficients) of orderk.Finally the trapezoidal-rule is shown to be asymptotically optimal for wide classes of Hilbert-spaces of periodic functions.

     

A dual algorithm for convex-concave data smoothing by cubicC 2-splines

Summary In this paper the problem of smoothing a given data set by cubicC ²-splines is discussed. The spline may required to be convex in some parts of the domain and concave in other parts. Application of splines has the advantage that the smoothing problem is easily discretized. Moreover, the special structure of the arising finite dimensional convex program allows a dualization such that the resulting concave dual program is unconstrained. Therefore the latter program is treated numerically much more easier than the original program. Further, the validity of a return-formula is of importance by which a minimizer of the orginal program is obtained from a maximizer of the dual program.The theoretical background of this general approach is discussed and, above all, the details for applying the strategy to the present smoothing problem are elaborated. Also some numerical tests are presented.     

Comparison theorems forL-monosplines of minimal norm

Summary LetLM _N be the set of allL-monosplines withN free knots, prescribed by a pair (x;E) of pointsx = {x _i } ₁ ⁿ ,a <x ₁ < ... <x _n <b and an incidence matrixE = (e _ij ) _i=1 ⁿ , _r-1 ^j=0 with Denote byLM _N ^O the subset ofLM _N consisting of theL-monosplines withN simple knots (n=N). We prove that theL-monosplines of minimalL _p-norms inLM _N belong toLM _N ^O .The results are reformulated as comparison theorems for quadrature formulae.     

Spline approximations to spherically symmetric distributions

Summary We discuss the problem of approximating a functionf of the radial distancer in ^d on 0r< by a spline function of degreem withn (variable) knots. The spline is to be constructed so as to match the first 2n moments off. We show that if a solution exists, it can be obtained from ann-point Gauss-Christoffel quadrature formula relative to an appropriate moment functional or, iff is suitably restricted, relative to a measure, both depending onf. The moment functional and the measure may or may not be positive definite. Pointwise convergence is discussed asn. Examples are given including distributions from statistical mechanics.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561.     

Optimal addition of knots to cubature formulae for planar regions

Summary A method is described to add knots to a cubature formula of degree 2k–1 for an integral over a symmetric region, to obtain a cubature formula of degree 2k+1. This method is used to construct cubature formulae for the square, the circle, the hexagon and the entire plane.     

A General framework for local interpolation

Summary We present a general framework for the construction of local interpolation methods with a given approximation order. Some applications to multivariate spline spaces are presented.Supported by the National Science Foundation, Contract Nos. DMS-8602337 and DMS-8701190Sponsored by Defense Advanced Research Projects Agency (DARPA), under contract No. MDA 972-88-C-0047 for DARPA Initiative in Concurrent Engineering (DICE)     

An a posteriori error estimation of the Rayleigh-Ritz eigenvalues

Summary LetA, B be essentially self-adjoint and positive definite differential operators defined inL ₂(G). Using Svirskij's construction of the base operator and some results from the analytic perturbation theory of linear operators a formula providing eigenvalue lower bounds of the problemAu=Bu is derived. In this formula a rough lower bound of some higher eigenvalue and the residual convergence of the Rayleigh-Ritz eigenfunction approximations are needed. Some numerical results are presented.     

An algorithm for computing constrained smoothing spline functions

Summary The problem of computing constrained spline functions, both for ideal data and noisy data, is considered. Two types of constriints are treated, namely convexity and convexity together with monotonity. A characterization result for constrained smoothing splines is derived. Based on this result a Newton-type algorithm is defined for computing the constrained spline function. Thereby it is possible to apply the constraints over a whole interval rather than at a discrete set of points. Results from numerical experiments are included.     

Strict Chebyshev approximation for general systems of linear equations

Summary An ascent exchange algorithm for computing the strict Chebyshev solution to general systems of linear equations is presented. It uses generalized exchange rules to ensure convergence and splits up the entire system into subsystems by means of a canonical decomposition of a matrix obtained by Gaussian elimination methods. All updating procedures are developed and several numerical examples illustrate the efficiency of the algorithm.     

Convergence of product formulas for the numerical evaluation of certain two-dimensional Cauchy principal value integrals

Summary We consider product rules of interpolatory type for the numerical approximation of certain two-dimensional Cauchy principal value integrals. We present convergence results which generalize those known in the one-dimensional case.Work sponsored by the Ministero della Pubblica Istruzione of Italy     

Estimation of the effect of numerical integration in finite element eigenvalue approximation

Summary Finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.The work of this author was partially supported by the National Science Foundation under Grant DMS-84-10324     

A numerical method to boundary value problems for second order delay differential equations

Summary In the first part of this paper we are dealing with theoretical statements and conditions which lead to existence and uniqueness of the solution of a nonlinear boundary value problem with delay. Next we apply this method successfully to a numerical example. The computations have been carried out at the computer Siemens 4004. The data obtained are presented in two tables.     

On a problem proposed by H. Braß concerning the remainder term in quadrature for convex functions

Summary We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw1, is of best possible orderO(n ^–2) for every bounded convex function. This result solves an open problem proposed by H. Braß and published in the problem section of the proceedings of the 2. Conference on Numerical Integration held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hämmerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw _, (x)=(1–x)(1+x), we show that the above result for Gaussian quadrature is not valid precisely ifw _, is unbounded.Dedicated to Prof. H. Braß on the occasion of his 55th birthday     

An algorithm for best approximate solutions ofAx=b with a smooth strictly convex norm

Summary In this paper, overdetermined systems ofm linear equations inn unknowns are considered. With ^m equipped with a smooth strictly convex norm, ·, an iterative algorithm for finding the best approximate solution of the linear system which minimizes the ·-error is given. The convergence of the algorithm is established and numerical results are presented for the case when · is anl _p norm, 1<p<.Portions of this paper are taken from the author's Ph.D. thesis at Michigan State University     

Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.