Showing papers in "Mathematics of Computation in 1989"

TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework

Abstract: Construction et analyse d'une classe de methodes a elements finis de Galerkin discontinues a variation totale bornee pour la resolution des lois de conservation. Etude de la convergence. Resultats numeriques

Inverse Problems in Vibration

Abstract: This article reviews recent literature on inverse problems relating to the reconstruction or estimation of the physical properties of mechanical systems from a knowledge of (some of) their spectral and/or modal data. The review relates exclusively to small vibrations of mechanical systems. The literature is divided according to: the type of system, continuous or discrete, damped or undamped; the type of data, spectral, modal, nodal, complete or incomplete.

An absolutely stabilized finite element method for the stokes problem

Abstract: On presente une formulation a elements finis absolument stabilisee pour le probleme de Stokes On etablit des estimations d'erreurs optimales dans la norme L 2 pour l'approximation des champs de vitesses et de pressions

The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity

Abstract: A particle method for convection-diffusion equations based on the approximation of diffusion operators by integral operators and the use of a particle method to solve integro-differential equations previously described is presented and studied. The isotropic diffusion operators are dealt with first. Two approximation possibilities are obtained, depending on whether or not the integral operator is positive. An extension of the method to anisotropic diffusion operators follows. The consistency and the accuracy of the method require much more complex conditions on the cutoff functions than in the isotropic case. After detailing these conditions, several examples of cutoff functions which can be used for practical computations are given. A detailed error analysis is then performed. 24 refs.

Finite element-galerkin approximation of the eigenvalues and Eigenvectors of selfadjoint problems

Abstract: : This paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or, more generally, Galerkin methods. Suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let M(lambda) denote the space of eigenvectors corresponding to lambda. Keywords: Hilbert space.

A Finite Difference Domain Decomposition Algorithm for Numerical Solution of the Heat Equation

Abstract: A domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented. In this procedure, interface values between subdomains are found by an explicit finite difference formula. Once these values are calculated, interior values are determined by backward differencing in time. A natural extension of this method allows for the use of different time steps in different subdomains. Maximum norm error estimates for these procedures are derived, which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters.

Nonnegativity-, monotonicity-, or convexity-preserving cubic and quintic Hermite interpolation

Abstract: The Hermite polynomials are simple, effective interpolants of discrete data. These interpolants can preserve local positivity, monotonicity, and convexity of the data if we restrict their derivatives to satisfy constraints at the data points. This paper de- scribes the conditions that must be satisfied for cubic and quintic Hermite interpolants to preserve these properties when they exist in the discrete data. We construct algorithms to ensure that these constraints are satisfied and give numerical examples to illustrate the effectiveness of the algorithms on locally smooth and rough data. 1. Introduction. Piecewise polynomial interpolants, especially those based on Hermite polynomials (polynomials determined by their values and values of one or more derivatives at both ends of an interval), have a number of desirable properties. They are easy to compute once the derivative values are chosen. If the derivative values are chosen locally (e.g., by finite difference methods), then the interpolant at a given point will depend only on the given data at nearby mesh points. If the derivatives are computed by spline methods, then the interpolant will have an extra degree of continuity at the mesh points. In either case, the interpolant is linear in the given function values and has excellent convergence properties as the mesh spacing decreases. These methods, however, do not necessarily preserve the shape of the given data. When the data arise from a physical experiment, it may be vital that the interpolant preserve nonnegativity (f(x) > 0), nonpositivity (f(x) 0 or f(x) 0), or concavity (f(x) < 0). In this and other cases, geometric considerations, such as preventing spurious behavior near rapid changes in the data, may be more important than the asymptotic accuracy of the interpolation method. One can construct a shape-preserving interpolant by constraining the derivatives for the Hermite polynomials to meet conditions which imply the desired properties ((4), (5), (8), (11)—(15), (20)), by adding new mesh points

Continuous finite elements in space and time for the heat equation

Abstract: On analyse une nouvelle methode variationnelle pour l'approximation de l'equation de la chaleur a l'aide des elements finis continus dans l'espace et le temps. On presente les estimations d'erreurs et les resultats d'experiences numeriques

Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions

Abstract: Convergence of a shock-capturing streamline diffusion finite element method for a conservation law in two space dimensions

An Optimal-Order Multigrid Method for P1 Nonconforming Finite Elements

Abstract: Etude d'une methode multigrille d'ordre optimal pour la resolution des problemes aux limites elliptiques du second ordre a l'aide d'elements finis non conformes

The weighted particle method for convection-diffusion equations. II. The anisotropic case

Abstract: On etend la methode particulaire, utilisee dans la partie I pour les equations de convection-diffusion, aux operateurs de diffusion anisotrope. Des conditions plus complexes que dans le cas isotrope sont exigees pour les fonctions de cutoff. Une analyse d'erreur detaillee est effectuee

Nonlinear filters for efficient shock computation

Abstract: A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

Error estimates for semidiscrete finite element methods for parabolic integro-differential equations

Abstract: Application des methodes d'elements finis aux equations integro-differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d'ordre β≤2

Computers and Mathematics

Abstract: : Computers and Mathematics '89 is the third in a series of conferences devoted to the use of computers in mathematics and the mathematical sciences This is interpreted in a broad sense; computers are used in mathematics not just for approximate numerical calculation but in virtually every area of pure and applied mathematics, including algebra, geometry, number theory, group theory, integration and differential equations Each of the conferences in this series has had a strong, interdisciplinary program of invited speakers In Computers and Mathematics '89 the contributed research papers have assumed an equally important role This volume contains the contributed papers accepted for presentation, selected from 85 drafts submitted in response to the call for papers Partial contents: Summation of Harmonic Numbers; Finite-Basis Theorems and a Computation-Integrated Approach; Practical Determination of the Dimension of an Algebraic Variety; A Computer Generated Census of Cusped Hyperbolic 3- Manifolds; Symmetric Matrices with Alternating Blocks; Use of Symbolic Methods in Analyzing an Integral Operator; Computer Algebraic Methods for Investigating Plane Differential Systems of Center and Focus Type; An Example of Computer Enhanced Analysis

Essentially nonoscillatory spectral Fourier methods for shock wave calculations

Abstract: In this paper, we present an essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.

Nonconforming finite elements for the Stokes problem

Abstract: On obtient un nouveau resultat de stabilite pour l'approximation du probleme de Stokes stationnaire avec des elements finis non conformes

Superconvergence for multistep collocation

Abstract: Application des considerations de collocation aux methodes multiples. Etude de la superconvergence

On the Convergence of the p-Version of the Boundary Element Galerkin Method.

Abstract: : The authors consider various physical problems which may be formulated in terms of integral equations of the first kind, including the two-dimensional screen Neumann and Dirichlet problems in acoustics (and crack problems in elasticity) Sharp regularity results for the solutions are available for these problems Proven is the convergence of the p-version for some Galerkin boundary element schemes based on the integral equation formulations It is shown that the rate of convergence obtained by our method is twice that for the usual h-version

Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems

Abstract: Etude des proprietes de stabilite et de convergence de la methode des directions alternees implicite losrsqu'elle est appliquee aux problemes aux valeurs initiales dans un espace bidimensionnel

Comment on Calderon's Paper: "On an Inverse Boundary Value Problem"

Abstract: Calder6n determined a method to approximate the conductivity a of a con-. ducting body in Rn (for n > 2) based on measurements of boundary data. The approximation is good in the L. norm provided that the conductivity is a small perturbation from a constant. We calculate the approximation exactly for the case of homogeneous concentric conducting disks in R2 with different conductivities. Here, the difference in the conductivities is the perturbation. We show that the approximation yields precise information about the spatial variation of a, even when the perturbation is large. This ability to distinguish spatial regions with different conductivities is important for clinical monitoring applications.

The representation of lattice quadrature rules as multiple sums

Abstract: On donne une classification des regles de quadrature reticulees. On etablit des theoremes de representation

Improved condition number for spectral methods

Abstract: Introduction d'une methode spectrale avec un nombre de conditionnement O(N 2 ), N etant le degre maximum des polynomes