ERROR-ESTIMATES WITH SHARP CONSTANTS FOR A FADING MEMORY VOLTERRA PROBLEM IN LINEAR SOLID VISCOELASTICITY

The problem characterizing nonageing linear isothermal quasi-static is otropic compressible solid viscoelasticity in the time interval [0, T] is described. This is essentially a Volterra equation of the second k ind arrived at by adding smooth fading memory to the elliptic linear e lasticity equations. We analyze the errors resulting from replacing th e relaxation functions with practical approximations, in a semidiscret e finite element approximation, and in a fully discrete scheme derived by replacing the hereditary integral with the trapezoidal rule for nu merical integration. The error estimates are sharp in the sense that i f certain bounds on the data are independent of T, then so also are th e constants involved in them. This is a consequence of bypassing the u sual Gronwall lemmas with arguments that are more sensitive to the fad ing memory of the physical problem.