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

Abstract. We give a space-time Galerkin nite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading mem-ory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree (cG( )), with a discontinuous Galerkin piecewise con-stant (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 exible: 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. Key words. Viscoelasticity, adaptivity, nite element method, discontinuous Galerkin method, a pos-teriori error estimates. AMS subject classications. 73F15, 45D05, 65M60 1. Introduction. Th