Improving the accuracy of the mini-element approximation to Navier-Stokes equations

^** Email: blanca{at}imati.cnr.it^*** Email: frutos{at}mac.cie.uva.es^**** Corresponding author. Email: julia.novo{at}uam.es A technique to improve the accuracy of the mini-element approximationto incompressible the Navier–Stokes equations is introduced.Once the mini-element approximation has been computed at a fixedtime, the linear part of this approximation is postprocessedby solving a discrete Stokes problem. The bubble functions neededto stabilize the approximation to the Navier–Stokes equationsare not used at the postprocessing step. This postprocessingprocedure allows us to increase by one unit (up to a logarithmicterm) the H¹ norm rate of convergence of the velocity and correspondinglythe L² norm of the pressure. An error analysis of the algorithmis performed.     

Optimal control for cooperative parabolic systems governed by Schrodinger operator with control constraints

^** Email: Bahaa_gm{at}hotmail.com A distributed control problem for cooperative parabolic systemsgoverned by Schrödinger operator is considered. The performanceindex is more general than the quadratic one and has an integralform. Constraints on controls are imposed. Making use of theDubovitskii–Milyutin theorem given by Walczak (1984, Onesome control problems. Acta Univ. Lod. Folia Math., 1, 187–196),the optimality conditions are derived for the Dirichlet problem.     

An Efficient Algorithm for Solving the Rectilinear Multifacility Location Problem

One of the efficient methods for solving large rectilinear multifacilitylocation problems is the Direct Search method. The only drawbackof this method lies in the following difficulty. In some situations,when t new facilities are located together at one point, thenumber of arithmetic operations which are needed to establishoptimality is proportional to t²2^t. Therefore the method needsa prohibitive amount of computation time whenever t exceeds,say, 20. This paper gives a simple remedy for this problem.The paper states and proves a new necessary and sufficient optimalitycondition. This condition transforms the problem of computinga descent direction into a constrained linear least-squaresproblem. The latter problem is solved by a relaxation methodthat takes advantage of its special structure. The new techniqueis incorporated into the Direct Search method. This yields animproved algorithm that handles efficiently very large clusters.Numerical results are included.     

Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems

An approach via inertial-manifold theory is presented as a wayto study the problem of stabilizing semilinear diffusion systemsusing finite-dimensional controllers. It is shown that a Sakawatype of controller plays an important role in the constructionof an inertial manifold for the closed-loop (controlled) semilineardiffusion system. This means that the use of a Sakawa type ofcontroller reduces the stabilization problem for the closed-loopsystem to the one on the inertial manifold. ^*This paper was partially presented at the 12th IFAC World Congress,Sydney, 18–23 July 1993.     

Sharp threshold of global existence for non-linear Gross-Pitaevskii equation in RN

^** Email: chenguanggan{at}hotmail.com This paper is concerned with the non-linear Gross–Pitaevskiiequation which describes the attractive Bose–Einsteincondensate under a magnetic trap. By an intricate variationalargument we derive out a sharp threshold of blowing up and globalexistence by applying the potential well argument and the concavitymethod. Furthermore, we answer the question: How small are theinitial data, the global solutions of the Cauchy problem ofthe equation exist for [graphic: see PDF]     

Saddle points for maximin investment problems with observable but non-predictable parameters: solution via heat equation{dagger}

^** Email: nikolai.dokuchaev{at}ul.ie We study optimal investment problem for a market model wherethe evolution of risky assets prices is described by Itô'sequations. The risk-free rate, the appreciation rates and thevolatility of the stocks are all random; they depend on a randomparameter that is not adapted to the driving Brownian motion.The distribution of this parameter is unknown. The optimal investmentproblem is stated in a ‘maximin’ setting to ensurethat a strategy is found such that the minimum of expected utilityover all possible distributions of parameters is maximal. Weshow that a saddle point exists and can be found via a solutionof the standard 1D heat equation with a Cauchy condition definedvia one dimensional minimization. This solution even coversmodels with unknown solution for a given distribution of themarket parameters.     

Reimbursement systems and regional inpatient allocation: a non-linear optimisation model

^** Email: marion.rauner{at}univie.ac.at^*** Email: georg.schneider{at}univie.ac.at^**** Email: kurt.heidenberger{at}univie.ac.at This study presents a non-linear optimisation model for investigatingthe optimal allocation of both budgets and inpatients with differenttreatments among hospitals within a geographic region such asVienna. The objective function maximises the overall qualityof treatment provided by regional hospitals. We compare theeffects of two different reimbursement systems—fixed versusvariable budgets—on optimal allocation strategies. Thecombination of modelling ideas from hospital location-allocationmodels and economic models to solve such a problem is new accordingto the literature. We found that fixed budgets outperformedvariable budgets as fewer Euros had to be invested for an incrementalunit of quality of care provided in most of the policy scenariosanalysed. Regional demand and supply patterns for differenttreatments affect the decision makers' choice of the most suitablereimbursement system. In our illustrative example, two hospitalsappeared inefficient regardless of the reimbursement system.Vienna policy makers are currently considering restructuringthese hospitals. They plan to merge one with nearby hospitalsand transform the other into a nursing home.     

Hedging entry and exit decisions: optimizing location under exchange rate uncertainty

^** Email: alexru00{at}ms41.hinet.net^*** Email: ctlin{at}mail.yust.edu.tw The Cobb–Douglas production function with Abel's (1983,Am. Econ. Rev., 173, 228–233) model is extended herein,and real options analysis (ROA) for entry–exit decision-makingestablished utilizing Dixit's (1989b) decision model under exchangerate uncertainty. This work considers the effects of real exchangerates on strategies that determine the locations of productionby firms that are entering markets in two countries. The ROAis also adopted to evaluate the switching location between twocountries. A continuous-time model optimization problem is solvedin closed-form. This provides a useful beginning to an importantanalysis of the effects on industry of exchange rate fluctuationswhen the optimal entry (exit) trigger for transferring locationsis important for a basic global logistics model. Furthermore,a myopic solution of the optimal entry (exit) trigger, sensitivityanalysis and some characteristics of the optimal productionstrategy are sought. This paper contributes to the problem ofchoice of foreign production strategy.     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

On the numerical solution of involutive ordinary differential systems

We consider a Galerkin finite element method that uses piecewisebilinears on a modified Shishkin mesh for a model singularlyperturbed convection–diffusion problem on the unit square.The method is shown to be convergent, uniformly in the perturbationparameter , of order N^–1in a global energy norm, providedonly that N^–1, where O(N²)mesh points are used. Thuson the new mesh the method yields more accurate results thanon Shishkin’s original piecewise uniform mesh, whereit is convergent of order N^–1lnN. Numerical experiments support our theoretical results. Received 14 September, 1998. Revised 24 September, 1999.     

Asymptotic expansions of the generalized Epstein-Hubbell integral

The generalized Epstein–Hubbell integral recently introducedby Kalla & Tuan (Comput. Math. Applic. 32, 1996) is consideredfor values of the variable k close to its upper limit k = 1.Distributional approach is used for deriving two convergentexpansions of this integral in increasing powers of 1 –k². For certain values of the parameters, one of these expansionsinvolves also a logarithmic term in the asymptotic variable1 – k². Coefficients of these expansions are given interms of the Appell function and its derivative. All the expansionsare accompanied by an error bound at any order of the approximation.Numerical experiments show that this bound is considerably accurate.     

Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem

We consider a Galerkin finite element method that uses piecewisebilinears on a modified Shishkin mesh for a model singularlyperturbed convection–diffusion problem on the unit square.The method is shown to be convergent, uniformly in the perturbationparameter , of order N^–1in a global energy norm, providedonly that N^–1, where O(N²)mesh points are used. Thuson the new mesh the method yields more accurate results thanon Shishkin’s original piecewise uniform mesh, whereit is convergent of order N^–1lnN. Numerical experiments support our theoretical results. Received 14 September, 1998. Revised 24 September, 1999.     

An algebraic analysis approach to linear time-varying systems

^** Email: zerz{at}mathematik.uni-kl.de This paper introduces an algebraic analysis approach to lineartime-varying systems. The analysis is carried out in an ‘almosteverywhere’ setting, i.e. the considered signals are smoothexcept for a set of measure zero, and the coefficients of thelinear ordinary differential equations are supposed to be rationalor meromorphic functions. The methodology is based on a normalform for matrices over the resulting ring of differential operators,which is a non-commutative analogue of the Smith form. Thisis used to establish a duality between linear time-varying systemson the one hand and modules over the ring of differential operatorson the other. This correspondence is based on the fact thatthe signal space is an injective cogenerator when consideredas a module over this ring of differential operators.     

Minimal bases of matrix pencils and coprime matrix fraction descriptions

Two new methods for the construction of coprime MFDs are proposedstarting from a minimal state space realization of a transferfunction matrix. The approaches are based on the constructionof minimal bases for the kernels of the state-space-based pencils[s I – A, – B], [s I – A^t, –C^t]. Thelatter problem is explicitly solved using the Toeplitz matrixconstruction of minimal bases of matrix pencils. The numericalalgorithms of the methods are presented and some examples demonstratingthe implementation of the algorithms are also given. Received 18 May 2000. Revised 5 September 2000. Accepted 5 September 2000.     

New perturbation analyses for the Cholesky factorization

We present new perturbation analyses for the Cholesky factorizationA = R^T R of a symmetric positive definite matrix A. The analysesmore accurately reflect the sensitivity of the problem thanprevious normwise results. The condition numbers here are alteredby any symmetric pivoting used in PAP^T = R^TR, and both numericalresults and an analysis show that the standard method of pivotingis optimal in that it usually leads to a condition number veryclose to its lower limit for any given A. It follows that thecomputed R will probably have greatest accuracy when we usethe standard symmetric pivoting strategy. Initially we give a thorogh analysis to obtain both first-orderand strict normwise perturbation bounds which are as tight aspossible, leading to a definition of an optimal condition numberfor the problem. Then we use this approach to obtain reasonablyclear first-order and strict componentwise perturbation bounds. We complete the work by giving a much simpler normwise analysiswhich provides a somewhat weaker bound, but which allows usto estimate the condition of the problem quite well with anefficient computation. This simpler analysis also shows whythe factorization is often less sensitive than we previouslythought, and adds further insight into why pivoting usuallygives such good results. We derive a useful upper bound on thecondition of the problem when we use pivoting. This research was supported by the Natural Sciences and EngineeringResearch Ciuncil of Canada Grant OGP0009236. This research was supported in part by the US National ScienceFoundation under grant CCR 95503126.     

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L₂ norm error estimator is proposed for thePoisson equation. The error estimator can be applied to anisotropictetrahedral or triangular finite element meshes. The estimatoris rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropicbubble functions and the corresponding inverse inequalities.The upper error bound utilizes non-standard anisotropic interpolationestimates. Its proof requires H² regularity of the Poisson problem,and its quality depends on how good the anisotropic mesh resolvesthe anisotropy of the problem. This is measured by a so-called‘matching function’. A numerical example supports the anisotropic error analysis.     

On the Algorithmic Complexity of Minkowski's Reconstruction Theorem

In 1903 Minkowski showed that, given pairwise different unitvectors µ₁, ..., µ_m in Euclidean n-space Rⁿ whichspan Rⁿ, and positive reals µ₁, ..., µ_m such that^m_i=1µ_iµ_i = 0, there exists a polytope P in Rⁿ, uniqueup to translation, with outer unit facet normals µ₁, ...,µ_m and corresponding facet volumes µ₁, ..., µ_m.This paper deals with the computational complexity of the underlyingreconstruction problem, to determine a presentation of P asthe intersection of its facet halfspaces. After a natural reformulationthat reflects the fact that the binary Turing-machine modelof computation is employed, it is shown that this reconstructionproblem can be solved in polynomial time when the dimensionis fixed but is #P-hard when the dimension is part of the input. The problem of ‘Minkowski reconstruction’ has variousapplications in image processing, and the underlying data structureis relevant for other algorithmic questions in computationalconvexity.     

Two-dimensional nonlinear advection-diffusion in a model of surfactant spreading on a thin liquid film

The spreading of a localized monolayer of dilute, insoluble surfactant, discharged from a point source that moves at constant speed over a thin liquid film coating a planar substrate, is described according to lubrication theory by a pair of coupled nonlinear evolution equations for the monolayer concentration and the film depth h. Numerical and asymptotic techniquesare here used to show that the extent and structure of sucha spreading asymmetric monolayer can be well approximated bya single nonlinear advection–diffusion equation involving alone. At large times the solution is composed of three, spatiallydistinct, asymptotic regions: (i) a quasi-steady ‘nose’region (containing the source), in which there is a dominantbalance between two-dimensional nonlinear diffusion and advection;(ii) an ‘advective’ region, in which longitudinaladvection balances transverse diffusion; and (iii) a ‘tail’region, in which unsteady diffusion is dominant. In each region,local similarity solutions are obtained either exactly (inthe advective region) or approximately (elsewhere) by rescalingnumerical solutions of the initial-value problem. If the sourceconcentration decreases with time, it is demonstrated that the monolayer’s width is greatest in the tail region, whereasfor a source of increasing concentration the monolayer is widestin the advective region. For the simpler one-dimensional problemof a monolayer spreading from a line source, the same balanceshold but with transverse diffusion eliminated; here self-similarsolutions are found in all three regions that agree closelywith numerical solutions of the initial-value problem. Received 7 October, 1998. Revised 11 April, 2000. ⁺ antoine@mip.ups-tlse.fr Present address: Division of Theoretical Mechanics, Schoolof Mathematical Sciences, University of Nottingham , UniversityPark, Nottingham NG7 2RD, UK. Oliver.Jensen@nottingham.ac.uk.     

The streamline-diffusion method for nonconforming Qrot1 elements on rectangular tensor-product meshes

When the streamline–diffusion finite element method isapplied to convection–diffusion problems using nonconformingtrial spaces, it has previously been observed that stabilityand convergence problems may occur. It has consequently beenproposed that certain jump terms should be added to the bilinearform to obtain the same stability and convergence behaviouras in the conforming case. The analysis in this paper showsthat for the Q^rot₁ 1 element on rectangular shape-regular tensor-productmeshes, no jump terms are needed to stabilize the method. Inthis case moreover, for smooth solutions we derive in the streamline–diffusionnorm convergence of order h^3/2 (uniformly in the diffusion coefficientof the problem), where h is the mesh diameter. (This estimateis already known for the conforming case.) Our analysis alsoshows that similar stability and convergence results fail tohold true for analogous piecewise linear nonconforming elements.     

Optimal Petrov--Galerkin Methods through Approximate Symmetrization

^*Present address: Department of Mathematics, Imperial College, London SW7 2BZ. A technique of approximate symmetrization is used to derivea test space from a given trial space for a Petrov—Galerkinmethod. This is applied to one-dimensional diffusion—convectionproblems to give approximations which are near optimal in anenergy norm. Rigorous and precise error bounds are derived todemonstrate the uniformly good behaviour and near optimalityof the procedure over all values of the mesh Péclet number.