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

Abstract

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

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

“…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].…”

“…In closing here we remark that estimates for u t , u tt , etc, as well as error estimates resulting from incorrect data can also be derived in the L p -energy norm thus generalizing Theorems 3.1 and 3.2 in [16].…”

“…Equation (1) represents an abstract formulation of the linear quasistatic compressible viscoelasticity problem, as described for example in either of [16] or [17], and can be thought of as a Volterra integral equation of the second kind taking values in the Hilbert space V . In this paper we will demonstrate that by making physically reasonable assumptions of fading memory on the function φ, it is possible to use some relatively simple results from convolution Volterra theory to derive optimal L p (J ; · ) data-stability estimates for u.…”

Negative norm error control for second-kind convolution Volterra equations

“…The implementation details of the numerical scheme represented by (5) are covered in [7]. Here we remark only that in the fully practical scheme it may be necessary to use quadrature replacements for the integrals, and above we used second-order replacements to ensure that the Galerkin error dominates.…”

“…This appears to be the first time that such error control is possible for (1) since adaptive solvers based on classical discretizations are often unreliable in that they may not be able to meet a given tolerance. See for example the collocation scheme given by Blom and Brunner in [2], the linear multistep method of Jones and McKee in [4], and Shaw, Warby and Whiteman's trapezoidal algorithm in [5]. The price we pay for reliability is error control in the less easily interpreted W −1 p (J ) norm.…”