Stability of a family of multi-time-level difference schemes for the advection equation

Summary For the linear advection equation we consider explicit multi-time-level schemes of highest order which are one step in space direction only. If a stencil involvesk time steps we show that it is stable in theL ₂-sense for Courant numbers in the interval (0, 1/k). Since the order is 2k–1 one can use these schemes for high order discretization of the boundary conditions in hyperbolic initial value problems.Part of this work has been performed in the project Mehrschritt-Differenzenschemata of the Schwerpunktprogramm Finite Approximationen in der Strömungsmechanik which has been supported by the DFG     

Stability radius of polynomials occurring in the numerical solution of initial value problems

This paper deals with polynomial approximations(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by stability requirements, we present a numerical study of the largest diskD()={z C: |z+|} that is contained in the stability regionS()={z C: |(z)|1}. The radius of this largest disk is denoted byr(), the stability radius. On the basis of our numerical study, several conjectures are made concerningr _m,p=sup {r(): _m,p}. Here _{m, p} (1pm; p, m integers) is the class of all polynomials(x) with real coefficients and degree m for which(x)=exp(x)+O(x ^p+1) (forx 0).     

High levelm-harmonic cardinal B-splines

We generalize the notion ofm-harmonic cardinal B-spline defined in [Rabut, [6c]] to obtain B-splines on an infinite regular grid, which are halfway between elementary B-splines and the Lagrangean cardinal spline function. We give the main properties of these functions: Fourier transform, decay when x , integration,P _k -reproduction (fork<-2m–1) of the associated B-spline approximation, etc. We show that, in some sense, high levelm-harmonic B-splines may be considered as a finer regular approximation of the Dirac distribution than the elementarym-harmonic B-splines are.     

Microstructure evolution model in micromagnetics

The proposed model combines tendency for minimization of Gibbs magnetic energy with the rate-independent maximum-dissipation mechanism that reflects the macroscopical quantity of energy required to change one pole of a magnet to another. The microstructure is described on a mesoscopical level in terms of Young measures. Such mesoscopical, distributed-parameter model is formulated (and, after a suitable regularization), analyzed, discretized, implemented, and eventually tested computationally on a uni-axial magnet. The desired hysteresis macroscopical response is demonstrated together with the influence of material properties.     

L 2-approximation by the translates of a function and related attenuation factors

Summary Let be thek-dimensional subspace spanned by the translates (·–2j/k),j=0, 1, ...,k–1, of a continuous, piecewise smooth, complexvalued, 2-periodic function . For a given functionfL ²(–, ), its least squares approximantS _kf from can be expressed in terms of an orthonormal basis. Iff is continuous,S _kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of include the space of trigonometric polynomials where is the de la Valleé Poussin kernel, algebraic polynomial splines where is the periodic B-spline, and trigonometric polynomial splines where is the trigonometric B-spline.     

Linear operators generated by stochastic processes

Let X_t(0 t < ) be a homogeneous stochastically continuous stochastic process with independent increments; (, A, P) be the corresponding probability space; p, q1 be fixed numbers. Necessary and sufficient conditions are found for a stochastic integral defined on finite-valued functions to be extendable to a bounded operator from L_p (0, ) to L_q() (p q).Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 7–13, 1988.     

Elementarym-harmonic cardinal B-splines

We generalize the notion of B-spline to the thin plate splines and to otherd-dimensional polyharmonic splines as defined in [Duchon, [3]]; for regular nets, we give the main properties of these B-splines: Fourier transform, decay when x , stability, integration property, links between B-splines of different orders or of different dimensions and in particular link with the polynomial B-splines, approximation using B-splines... We show that, in some sense, B-splines may be considered as a regularized form of the Dirac distribution.     

Quadratically constrained least squares and quadratic problems

Summary We consider the following problem: Compute a vectorx such that Ax–b₂=min, subject to the constraint x₂=. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse.It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx ^T A x–2b ^T x subject to the constraint x₂=.Some numerical examples are given.This work was in part supported by the National Science Foundation under Grant DCR-8412314 and by the National Institute of Standards and Technology under Grant 60NANB9D0908.     

Stream vectors in three dimensional aerodynamics

Summary This work deals with the decomposition of a vector fieldu intou=×+. Non homogeneous boundary conditions on or are investigated; applications to the computation of inviscid flows are given; finally a conforming finite element implementation is studied and tested.     

Finite element approximation of the volume-matching problem

Summary Optimal orderH ¹ andL error bounds are obtained for a continuous piecewise linear finite element approximation of the volume matching problem. This problem consists of minimising |v| _1, ² overvH ¹() subject to the inequality constraintv0 and a number of linear equality constraints. The presence of the equality constraints leads to Lagrange multipliers, which in turn lead to complications with the standard error analysis for variational inequalities. Finally we consider an algorithm for solving the resulting algebraic problem.Supported by a SERC research studentship     

On a problem of A. Pleijel

In 1955, Arne Pleijel proposed the following problem which remains unsolved to this day: Given a closed plane convex curve C and a point x() at a fixed distance above the plane, as the point x() varies, characterize the point for which the conical surface with vertex x() and base C attains its minimum, and determine the limits as 0 and of this minimum point. The purpose of this paper is to solve the cases where approach its extremities and in the course of the solution, we obtain an interesting characterization of the limit points, which we shall call the Pleijel points of C. A consequence is that the inner Pleijel point provides an upper bound for the isoperimetric defect of C. We also generalize the problem to higher dimensional spaces, and obtain the corresponding characterizations of the limiting points for convex surfaces.     

Relationship between tetrahedron shape measures

Tetrahedron shape measures are used for evaluating the quality of tetrahedra in finite element meshes. Three shape measures, theminimum solid angle _min theradius ratio , and themean ratio , are discussed in this paper. A new formula for the computation of a solid angle of tetrahedron is derived. For different shape measures andv (with values 1), we establish a relationship between andv of the form wherec ₀,c ₁,e ₀, ande ₁ are positive constants. This means that if one measure approaches zero for a poorly-shaped tetrahedron, so does the other. Combined with the property that each measure attains a maximum value only for the regular tetrahedron, this means that the shape measures are equivalent.This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Pointwise accuracy of a finite element method for nonlinear variational inequalities

Summary Consider the following quasilinear elliptic PDE, which is equivalent to a nonlinear variational inequality: –divF(u)+(u)f. Here is a singular maximal monotone graph and the nonlinear differential operator is only assumed to be monotone; surfaces of prescribed mean curvature over obstacles may thus be viewed as relevant examples. The numerical approximation proposed in this paper consists of combining continuous piecewise linear finite elements with a preliminary regularization of . The resulting scheme is shown to be quasi-optimally accurate inL . The underlying analysis makes use of both a topological technique and a sharpL ^p -duality argument.This work was partially supported by Consiglio Nazionale delle Ricerche of Italy while the author was in residence at the Istituto di Analisi Numerica del C.N.R. di Pavia     

Finite elements for parabolic equations backwards in time

Summary Several regularization methods for parabolic equations backwards in time together with the usual additional constraints for their solution are considered. The error of the regularization is estimated from above and below. For a boundary value problem in time-method, finite elements as well as a time discretization are introduced and the error with respect to the regularized solution is estimated, thus giving an overall error of the discrete regularized problem. The algorithm is tested in simple numerical examples.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Characterization of regular singular linear systems of difference equations

This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations atz=. We are concerned with the behavior of solutions near the pointz= (the only fixed singularity for difference equations). It is important to know whether a system of linear difference equations has a regular singularity or an irregular singularity. To a given system () we can assign a number , called the Moser's invariant of (), so that the system is regular singular if and only if 1. We shall develop an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system of difference equations to an irreducible form. The computation ot the number can be done explicitly from this irreducible form.     

Construction d'une paramétrixe pour des opérateurs différentiels hypoelliptiques marimaux sur un espace homogéne d'un groupe de Lie nilpotent gradué de rang 2

Soient G une alébre de Lie nilpotente stratifée de rang 2, une sous-algébre de G, _0, la représentation de G dans l'espace L ²( \ G) indiute par le caractére trivial C, P un opérateur homogène appartenant à l'algébre universelle enveloppante (complexifiée) U(G) tel que l'opérateur _0, (P) soit hypoelliptique maximal. Cet opérateur peut s'exprimer par une intégrale dépendant de la restriction du symbole p de P au sousensemble = G · décrit par les orbites des éléments de dans la représentation contragrédiente de G dans G ^*.Une algèbre de symboles définis sur est construite et permet de déterminer une paramétrixe de _0, (P); des résultats de réguralité de cet opérateur dans des espaces de Sobolev adaptés sont ensuite obtenus.     

A θ-stable approximations of abstract Cauchy problems

Summary We study the approximation of linear parabolic Cauchy problems by means of Galerkin methods in space andA -stable multistep schemes of arbitrary order in time. The error is evaluated in the norm ofL _t ² (H _x ¹ ) L _t (L _x ² ).     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Commutativity of operators on the lattice of existence varieties

In a previous paper the authors introduced seven complete congruences on the lattice _ev(I of e-varieties of regular semigroups of the form _P :U _P VPU=PV, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form _P :U _P VPU=PV, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence on _ev(I) and eachU_ev(I), the -class ofU is an interval [U ,U ] so that there is associated with each such congruence an idempotent operatorUU on _ev(I). This paper establishes numerous results concerning the commutativity of operators of this form.This work was supported in part by NSERC Grant 4044.