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We consider a finite-element-in-space, and quadrature-in-timediscretization of a compressible linear quasistatic viscoelasticity problem. The spatial discretization uses a discontinous Galerkin finite element method based on polynomials of degree r-termed DG(r)-and the time discretization uses a trapezoidal-rectangle rule approximation to the Volterra (history) integral. Both semi-and fully-discrete a priori error estimates are derived without recourse to Gronwall's inequality, and therefore the error bounds do not show exponential growth in time. Moreover, the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution (e.g. when using the standard P k polynomial basis on simplicies, or tensor product polynomials, Q k , on quadrilaterals). When this is not the case (e.g. using P k on quadrilaterals) the convergence rate is suboptimal in r but remains optimal in h. We also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation.

We consider the usual linear elastodynamics equations augmented with evolution equations for viscoelastic internal stresses. A fully discrete approximation is defined, based on a spatially symmetric or non-symmetric interior penalty discontinuous Galerkin finite element method, and a displacement-velocity centred difference time discretisation. An a priori error estimate is given but only the main ideas in the proof of the error estimate are reported here due to the large number of (mostly technical) estimates that are required. The full details are referenced to a technical report.

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