Negative norm error control for second-kind convolution Volterra equations

Shaw, Sylvie; Whiteman, J. R.

doi:10.1007/pl00005390

Cited by 9 publications

(1 citation statement)

References 8 publications

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“…However, engineers still regard the quasistatic problem as an important model (see for example the beam problem described in [11]), and to solve practical design problems with complex geometries they of course need reliable adaptive software for numerical solution. It is our aim to provide such reliable numerical solution algorithms, built on rigorous a posteriori error estimates, and our first results are in [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

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.

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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.

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Section: Introductionmentioning

confidence: 99%

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.

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Modelling and Finite Element Analysis of Applied Polymer Viscoelasticity Problems

Shaw¹,

Johnson²,

Whiteman³

2000

The Mathematics of Finite Elements and Applications X

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A posteriori error estimates for space–time finite element approximation of quasistatic hereditary linear viscoelasticity problems

Shaw

Whiteman

2004

Computer Methods in Applied Mechanics and Engineering

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Abstract. We give a space-time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree, with a discontinuous Galerkin piecewise constant (dG(¥ )) or linear (dG( § )) approximation in time.A posteriori Galerkin-error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible: strong© -energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are nonsmooth (in time).We also give upper bounds on the dG(¥ )cG( § ) a posteriori error estimates which indicate optimality. However, a complete analysis is not given.

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Negative norm error control for second-kind convolution Volterra equations

Cited by 9 publications

References 8 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

Modelling and Finite Element Analysis of Applied Polymer Viscoelasticity Problems

A posteriori error estimates for space–time finite element approximation of quasistatic hereditary linear viscoelasticity problems

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