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.