Optimal long-time L-p (0,T) stability and semidiscrete error estimates forthe Volterra formulation of the linear quasistatic viscoelasticity problem

The purpose of this article is to show how the solution of the linear quasi static (compressible) viscoelasticity problem, written in Volterra form wit h fading memory, may be sharply bounded in terms of the data if certain phy sically reasonable assumptions are satisfied. The bounds are derived by mak ing precise assumptions on the memory term which then make it possible to a void the Gronwall inequality, and use instead a comparison theorem which is more sensitive to the physics of the problem. Once the data-stability esti mates are established we apply the technique also to deriving a priori erro r bounds for semidiscrete finite element approximations. Our bounds are der ived for viscoelastic solids and fluids under the small strain assumption i n terms of the eigenvalues of a certain matrix derived from the stress rela xation tenser. For isotropic materials we can be explicit about the form of these bounds, while for the general case we give a formula for their compu tation.