Pointwise error estimates for a streamline diffusion scheme on a Shishkin mesh for a convection-diffusion problem

We analyze a streamline diffusion scheme on a special piecewiseunform mesh for a model time-dependent convection–diffusionproblem. The method with piecewise linear elements is shownto be convergent, independently of the diffusion parameter,with a pointwise accuracy of almost order 5/4 outside the boundarylayer and almost order 3/4 inside the boundary layer. Numericalresults are also given. stynes{at}bureau.ucc.ie     

Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. II: convection-diffusion-type problems

e-mail address: stynes{at}bureau.ucc.ie We consider singularly perturbed high-order elliptic two-pointboundary value problems of convection-diffusion type. Undersuitable hypotheses, the coercivity of the associated bilinearform is proved and a representation result for the solutionsof such problems is given. A family of Galerkin finite-elementmethods based on piecewise polynomial test/trial functions ona Shishkin mesh is constructed and proved to be convergent,uniformly in the perturbation parameter, in energy and Wnorms. Numerical results are presentedfor a second-order problem and fourth-order problems.     

A numerical method using the Prfer transformation for the calculation of eigenpairs of two-parameter Sturm-Liouville problems

A shooting method for coupled Prfer equations is discussedfor numerical solution of two-parameter Sturm-Liouville problems. Research supported in part by grants from the NSERC of Canada.     

Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems

The long-time behaviour of continuous time Galerkin (CTG) approximationsof some well-known one-dimensional non-linear evolution problemswhich model phase transitions are analyzed. These numericalschemes are fully discrete and of arbitrary order. Partially supported by the Army Research Office through grant28535-MA. Partially supported by the ONR Contract No N00014-90-J-1238.     

Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws

The theory of optimal recovery is applied to finite volume methodsfor the numerical solution of conservation laws in multiplespace dimensions. Classical polynomial EN0 (essentially non-oscillatory)algorithms can be interpreted as trivial recovery operatorsfor the point functional in the light of this theory. Thin platesplines are identified as optimal recovery functions in Beppo-Levispaces and can be seen as multi-dimensional analogues of cubicsplines. Recovery algorithms of ENO-type based on thin platesplines are developed and applied to test problems includingthe Euler equations of compressible gas dynamics. E-mail: Thomas.Sonar{at}ts.go.dlr.de     

On the Convergence of Some Approximate Methods of Conformal Mapping

A method was given in Ellacott (1978) for determining approximatelythe conformal mapping of a Jordan region on to a disc. Someresults on the convergence of this method are given, which canbe used to prove the result (conjectured in Ellacott, 1978)that if the boundary curve is analytic, then convergence isuniform. The corresponding result is also proved for the Bergmanand Szeg Kernel methods with polynomial basis functions. (Theresult is already known for the Szeg? Kernel method, but a differentproof is given.) Also discussed is the use of rational basisfunctions for the Bergman Kernel method. Currently visiting Forschungsinstitut fr Mathematik, ETH-Zentrum,CH-8092, Zurich.     

Estimates of the region of attraction of continuous-time cascade systems

Xiao-Song Yang Department of Mathematics, Huazhong University of Science and Technology Wuhan, Hubei 430074, People's Republic of China Huimin Li Departments of Mathematics and Control and Engineering, Huazhong University of Science and Technology Wuhan, Hubei 430074, People's Republic of China Corresponding author. Email: yangxs{at}cqupt.edu.cn Received on May 1, 2006; Accepted on October 6, 2006 In this paper, we first present some new sufficient conditionsfor global asymptotic stability of continuous-time cascade systems.Then, we give some detailed results on the region of attractionof continuous-time cascade systems. Some examples are presentedto illustrate these results.     

A finite-difference algorithm for an inverse Sturm-Liouville problem

We study a method for approximating a potential q(x) in y(0)=y()=0 from finite spectral data. When the potential is symmetric,the data are the first M Dirichlet eigenvalues. In the generalcase, the first M terminal velocities are also specified. Acentred finite-difference scheme reduces the inverse Sturm-Liouvilleproblem to a matrix inverse eigenvalue problem. Our approachis motivated by the work of Paine, de Hoog and Anderssen, whoinvestigated the discrepancy between continuous and matrix eigenvaluesunder finite differences. Our modified Newton scheme is basedon choosing the number of interior mesh points in the discretizationto be 2M. The modified Newton scheme is shown to be convergentfor both the case of a symmetric and general potential. Somenumerical experiments are given. Supported in part by Institute for Scientific Computation,Texas A&M University.     

Stability of the approximation of a regular solution branch

In this paper we extend the so-called inf-sup conditions tononlinear problems having regular solution branches. The inf-supconditions for nonlinear problems have been introduced, forinstance, by Caloz and Rappaz in 1994. Here we present a moregeneral approach to include turning points on a solution branch.Our abstract results are applied to model examples. E-mail: Gabriel.Caloz{at}univ-rennesl.fr     

Simpler Polynomial Solutions in Control and Filtering

Kuera's completely algebraic method is used to derive a polynomialsolution with only one Diophantine equation for least-squaresdeterministic control in discrete-time. The same method is used,again involving only one Diophantine equation, to derive themultivariable discrete-time Wiener filter. This paper was presented at the IMA Symposium on Control Theory,Somerville College,Oxford, September 1986.     

Remarks on Perturbations of the Equation

Perturbations of the equation are discussed. The structure of near-orthogonal matrices is shown, and implicationsfor certain control problems are pointed out. Examples are givenand the circumstances are noted in which the real part of aneigenvalue can be expected to change sign. This paper was written while the author was on a Leave of Absenceat the Department of Mathematics, University of Salford.     

Minimal equivalence of differential IO systems

Differential algebraic controllability, observability, and minimalrealization are considered further, upon which a new equivalenceproblem, i.e. the minimal equivalence of two given differentialinput-output (IO) systems through state-space realization isconsidered. Some criteria for equivalence and minimal equivalenceof two IO systems are given. The algorithm for checking minimalequivalence of two IO systems with single input is implementedwith MACSYMA. Email: xyl{at}sun.engg.le.ac.uk David.Bell{at}umist.ac.uk     

On approximate approximations using Gaussian kernels

This paper discusses quasi-interpolation and interpolation withGaussians. Estimates are obtained showing a high-order approximationup to some saturation error negligible in numerical applications.The construction of local high-order quasi-interpolation formulasis given. Supported in part by the International Centre for MathematicalSciences, Edinburgh.     

Finite-difference schemes for scalar conservation laws with source terms

Explicit and semi-implicit finite-difference schemes approximatingnon-homogeneous scalar conservation laws are analyzed. Optimalerror bounds independent of the stiffness of the underlyingequation are presented. This author has been supported by The Norwegian Research Council(NFR), program No 100284/431. e-mail: schroll{at}igpm.rwth-aachen.de This author has been supported by The Norwegian Research Council(NFR), program Nos 100284/431 and STP.29643. e-mail: ragnar{at}ifi.uio.no     

Characterization of Distributions through Maximization of {alpha}-Entropies

This paper provides characterizations of the Pareto distributionof the second kind, the Pearson type-VII distribution and thet-distribution. These characterizations have been obtained throughmaximization of -entropies. As 1, characterizations of the exponential,the Laplace, and the normal distributions are obtained. On sabbatical leave from the Department of Statistics, YarmoukUniversity, Irbid, Jordan.     

An Example of Only Linear Convergence of Trust Region Algorithms for Non-smooth Optimization

Most superlinear convergence results about trust region algorithmsfor non-smooth optimization are dependent on the inactivityof trust region restrictions. An example is constructed to showthat it is possible that at every iteration the trust regionbound is active and the rate of convergence is only linear,though strict complementarity and second order sufficiency conditionsare satisfied. Presented at the 1983 Dundee Conference on Numerical Analysis     

Numerical simulations of measurements of capillary contact angles

In a recent paper, Fischer and Finn have proposed a procedureto improve the accuracy in the measurement of capillary contactangles, based on the use of vessels with canonical cross-sections.We simulate numerically the behaviour of such shapes for a numberof cross-sections and fluid contact angles. Our approximationconsists of the minimization of a suitable convex functionaldiscretized by finite elements. e-mail: bellettini{at}sns.it e-mail: paolini{at}isa.mat.unimi.it     

A Simple Proof for a Spectral Factorization Theorem

It is shown using simple methods that the transform Z(z), z C, of the coefficient sequence of the Wold decomposition ofany full-rank wide-sense stationary purely non-deterministicstochastic process satisfies (i) Z(z) H² (D) and (ii) Z^–1(z) H(D). Further it is shown that all spectral factors satisfying(i) and (ii) are equal up to right multiplication by orthogonalmatrices, and that among these the normalized (Z(0) =I) spectralfactors are equal to the transform of the Wold decomposition.An elementary proof of Youla's Theorem is then given togetherwith a simple proof that the rows of a Cholesky factor of abanded block Toeplitz matrix converge to the coefficients ofa stable matrix polynomial. Computer and Automation Institute of the Hungarian Academyof Sciences, Budapes, Hungary.     

Algebraic Manipulation of Polynomial Matrices: A Survey

A concise review is given of some recent methods for performingalgebraic operations on polynominal matrices. The problems addressedare division, inversion, and extraction of greatest common divisorof a pair of matrices. Based on a presentation at the International Symposium on theMathematical theory of Networks and Systems, Amsterdam, June1989.     

On the Stability Analysis of Nonlinear Systems

The stability of nonautonomous delay differential equationsis studied via a combination of the global linerization andnonlinear variation of parameter techniques. Permanent address: Department of Mathematics, University ofElectronics Science & Technology of China, Chengdu, China.