On Gronwall’s inequality

Chu, Sun-Chin; Metcalf, F. T.

doi:10.1090/s0002-9939-1967-0212529-0

Cited by 31 publications

(17 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although similar in spirit, the stability results presented below for viscoelastic solids are more general than those already given in [16], and, as we mentioned above, have application in the adaptive numerical solution of viscoelasticity problems represented by (1). Our results in this paper are even more general than in [16] in that we also derive stability estimates for viscoelastic fluids as considered by Golden and Graham in [8].…”

Section: B(t S)u(s) Dssupporting

confidence: 75%

See 1 more Smart Citation

Optimal long-time $L_p}(0,T)$ stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem

Shaw

Whiteman

2001

Numer. Math.

View full text Add to dashboard Cite

The purpose of this article is to show how the solution of the linear quasistatic (compressible) viscoelasticity problem, written in Volterra form with fading memory, may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied. The bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the Gronwall inequality, and use instead a comparison theorem which is more sensitive to the physics of the problem. Once the data-stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations. Our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor. For isotropic materials we can be explicit about the form of these bounds, while for the general case we give a formula for their computation.

show abstract

Section: B(t S)u(s) Dssupporting

confidence: 75%

“…Note that Lemma 5 may easily be used to prove Gronwall's inequality by taking φ = constant > 0. Similar comparison theorems can be found in the papers [1] and [4].…”

Section: Lemma 5 (Comparison Theorem) Forsupporting

confidence: 74%

Optimal long-time $L_p}(0,T)$ stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem

Shaw

Whiteman

2001

Numer. Math.

View full text Add to dashboard Cite

show abstract

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php given in [4] and [5, Thm. 2.1].…”

Section: Closurementioning

confidence: 99%

“…In the original version of this manuscript [16], these results were built on comparison theorems such as those given in [4] and [5,Thm. 2.1].…”

Section: Introductionmentioning

confidence: 99%

Error Estimates with Sharp Constants for a Fading Memory Volterra Problem in Linear Solid Viscoelasticity

Shaw¹,

Warby²,

Whiteman³

1997

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

The problem characterizing nonageing linear isothermal quasi-static isotropic compressible solid viscoelasticity in the time interval [0, T ] is described. This is essentially a Volterra equation of the second kind arrived at by adding smooth fading memory to the elliptic linear elasticity equations. We analyze the errors resulting from replacing the relaxation functions with practical approximations, in a semidiscrete finite element approximation, and in a fully discrete scheme derived by replacing the hereditary integral with the trapezoidal rule for numerical integration. The error estimates are sharp in the sense that if certain bounds on the data are independent of T , then so also are the constants involved in them. This is a consequence of bypassing the usual Gronwall lemmas with arguments that are more sensitive to the fading memory of the physical problem. Introduction.A priori stability and error estimates for time-dependent problems posed in the time interval [0, T ] often include a constant which is of the order e CT . Typically this is a consequence of applying Gronwall lemmas which do not always take proper account of any special information contained in the equations. In some cases it is known that the solution does not grow in this way. Such knowledge might, for example, derive from experience, exact solutions to model problems, or heuristic physical argument. In such cases it is arguable as to whether Gronwall-type estimates actually contain any useful information. Of course they can be used to prove convergence on finite time intervals, but this is only relevant as the time step approaches zero. This never happens in any practical computation. Rather, for longtime numerical integration it is important to know that the error is not exponentially increasing. Then and only then, it seems, can any quantitative meaning be attached to the computed results.For a specific problem in nonageing quasi-static linear isothermal isotropic compressible solid viscoelasticity we will show that sharp data-stability estimates (in the sense of the constants) can be derived as well as correspondingly sharp error estimates for particular discrete schemes. Using the traditional "Gronwall approach" this problem has already been analyzed in [15]. However, the estimate given there is pessimistic for solid viscoelasticity problems because experience shows (and we will demonstrate below) that the error is far better controlled than is indicated by the derived error bound. Note that we are not addressing the convergence rate implied by this estimate, only the constant.Before summarizing the contents of this article it is necessary to introduce some notation. Define R 0 := [0, ∞). For T ∈ R + := R 0 \ {0} we let J := [0, T ] represent a

show abstract

“…Note that i ≤ n, i a=1 m a = n and k n = i a=1 (−s a ) ma > 0. Since |e (n) (t)+k T | ≤ d, we have the following inequality according to Gronwall inequality [13], [14]: …”

Section: Appendix Proof Of Lemmamentioning

confidence: 99%

Theory and Substantiation of z0g1 Controller Conquering Singularity Problem of Output Tracking for a Class of Nonlinear System

Zhang

Wang

Yin

et al. 2014

2014 IEEE 17th International Conference on Computational Science and Engineering

View full text Add to dashboard Cite

The conventional controller for output tracking of a class of nonlinear system is not valid when encountering the singularity (i.e., division-by-zero) problem. To conquer the singularity problem, Zhang et al. have proposed a simple yet effective controller design method, i.e., aiding the conventional controller with gradient dynamics (GD), yielding a z0g1 controller in the framework of Zhang-gradient (ZG) method. As an indepth research, this paper provides the theory and proof on the performance of this z0g1 controller, which is exploited to conquer the singularity problem while fulfilling the trackingcontrol task. Besides, simulations and experiments substantiate and complement the theoretical analyses of such a z0g1 controller.

show abstract

On Gronwall’s inequality

Cited by 31 publications

References 6 publications

Optimal long-time $L_p}(0,T)$ stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem

Optimal long-time $L_p}(0,T)$ stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem

Error Estimates with Sharp Constants for a Fading Memory Volterra Problem in Linear Solid Viscoelasticity

Theory and Substantiation of z0g1 Controller Conquering Singularity Problem of Output Tracking for a Class of Nonlinear System

Contact Info

Product

Resources

About