Error Estimates For Approximate Solutions Research Articles

Difference methods for impulsive differential-functional equations

We consider a class of difference methods for initial value problems for first-order impulsive partial differential-functional equations. We give sufficient conditions for the convergence of a sequence of approximate solutions under the assumptions that the right-hand sides satisfy the nonlinear estimates of Perron type with respect to the functional argument. The proof of the stability of difference methods is based on a general theorem on the error estimate of approximate solutions for difference-functional equations of Volterra type with an unknown function of several variables.

The Comparison of Two Error Estimates for Approximate Solutions of the Poisson Equation

The purpose of the present paper is to discuss different error estimates for the numerical solution of a Dirichlet problem for the Poisson equation, calculated via the five point (discrete) Laplacian. Whereas the first error bound, a sum of ordinary L^\infty -moduli, is deduced from the usual stability inequality, use is made of some properties of the discrete Green function to verify another stability inequality in terms of l^1 -norms, which then implies the second estimate via \tau -moduli multiplied by a logarithm factor. In the following we will show that it depends on the solution of the boundary value problem which measure of smoothness or rather which error estimate delivers the correct rate of convergence. The paper concludes with some remarks illustrating relations to the well-known logarithm factor in connection with finite element approximation.

A Posteriori Error Estimates for Nonlinear Problems. Finite Element Discretizations of Elliptic Equations

We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.

A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.

On the Error Estimate of Approximate Solutions of the Discrete Volterra Equations of K‐th Order

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 71, Issue 7-8 p. 293-295 Short Communication On the Error Estimate of Approximate Solutions of the Discrete Volterra Equations of K-th Order† Prof. Dr. M Kwapisz, Prof. Dr. M Kwapisz University of Gdańsk, Department of Mathematics, ul. Wita Stwosza 57, PL-80-952 Godańsk, PolandSearch for more papers by this author Prof. Dr. M Kwapisz, Prof. Dr. M Kwapisz University of Gdańsk, Department of Mathematics, ul. Wita Stwosza 57, PL-80-952 Godańsk, PolandSearch for more papers by this author First published: 1991 https://doi.org/10.1002/zamm.19910710725 † Dedicated to the memory of LOTHAR COLLATZ AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume71, Issue7-81991Pages 293-295 RelatedInformation

An investigation of convergence and error estimate of approximate solution for a quasilinear parabolic integrodifferential equation

An investigation of convergence and error estimate of approximate solution for a quasilinear parabolic integrodifferential equation

Spectral method for Zakharov equation with periodic boundary conditions

This paper describes the spectral method for numerically solving Zakharov equation with periodic boundary conditions. This method is spectral method for spatial variable and difference method for time variable. We make error estimation of approximate solution and prove the convergence of spectral method. We had given the convergence rate. Also, we prove the stability of approximate method for initial values.

Error estimate of approximate solution for a quasilinear parabolic integrodifferential equation in the $L_p$-space

The Rothe-Galerkin method is used for discretization. The rate of convergence in $C(I, L_p(G))$ for the approximate solution of a quasilinear parabolic equation with a Volterra operator on the right-hand side is established.

On a method of structural analysis

In this paper we present the theoretical foundation concerning a famous method in structural analysis, the distribution of moment (or/and dispacement): the mathematical description, new criterion, convergence and error estimation of approximate solution and some possible generalizations to that of other constructions.

Lower norm error estimates for approximate solutions of differential equations with non-smooth coefficients

In this paper, we derive error estimates inLp-norm, 1≦p≦∞, for the ℒ2-Finite Element approximation to solutions of boundary value problems, where the coefficients are functions of bounded variation. The ℒ2-Finite Element Method was introduced in [3] and was shown to be effective for problems with non-smooth coefficient.

Error bounds via a new variational principle for classical Hamiltonian systems

An extremum variational principle for classical Hamiltonian systems [9] is used for error estimate of approximate solution for a mechanical system with one degree of freedom. The procedure is developed separately for two-point boundary value problems and for Cauchy problems. Finally, the theory is illustrated by two concrete problems.

ABOUT SOME NOTES IN THE VARIATIONAL PRINCIPLE

The bilinear from α(…) on the real Hilbert space is said to be coercive if there is α>0 such that (0.1)a(υ,υ)≥a‖υ‖H∀υ∈H.Assume that a (…) is a bilinear continous and coercive form on H. Then, giving a f υ H′ arbitrarily, There exists a unique u υ H such that a(u,υ)=f(υ). (0.2)Furthermore, u depends continuously on f. This is the Lax-Milgram Theorem obtained in 1954.In 1972, A.K. Aziz[l] improved the condition (0.1) and obtained the sufficient condition of Eq. (0.2) being Well-posed.In this paper, by different method, we proved, more simply, this sufficient condition in §1. Furthermore, we proved that this sufficient condition is also the necessary one. In §2 we improved the error estimate of approximate solution of (0.2). which was obtained by Céa(3) in 1964. In §3 we discussed the Well-posed problem of Eq.(2), when a(…) is monotonic. This result may be considered as a generalization of the dffinition of coercive, and the Lax-Milgram Theorem as a special example. In §4 we extended a result of variational inequality to the case of product space. This result include the theorem of variational inequality, which was proved by Lions(2) in 1967. Using the methodhere, it is possible to extend certain theorems of Guo Youzhongt(4).

ON THEORETICAL FOUNDATION CONCERNING DISTRIBUTION OF MOMENT

Here we present the theoretical foundation concerning the distribution of moment (or/and displacement): the mathematical description, convergence and error estimation of approximate solution and the generalization to that of net shell theory.

On Logarithmic Norms

This paper discusses some normlike properties of the logarithmic norm and related efficiency questions. Some computational questions are raised and some examples of use are indicated.

On nonlinear time-dependent multivelocity transport equations

The aim of this paper is to show the existence and uniqueness of a solution for a class of nonlinear multivelocity neutron transport equations and to present a method for the determination of the solution. A recursion formula for the construction of a solution by the method of successive approximations and an error estimate for approximate solutions are given. Both global and local solutions are discussed. The conditions imposed on the nonlinear equation are directly applicable to the linear multivelocity transport problem, and a recursion formula for this system is also given. In the process of approximations, only successive integrations are involved. This fact indicates a promising possibility for the calculation of numerical results by using a computer. A brief discussion of this possibility and the applicability of the results on nonlinear problems are included.

A quadratic programming method with error estimates for approximate solutions

A quadratic programming method with error estimates for approximate solutions

Odhad chyby při přibližném řešení integrálních rovnic

Odhad chyby při přibližném řešení integrálních rovnic