We prove error estimates for a finite element approximation of viscoelastic dynamics based on continuous Galerkin in space and time, both in energy norm and in $L^2$ norm. The proof is based on an error representation formula using a discrete dual problem and a stability estimate involving the kinetic, elastic, and viscoelastic energies. To set up the dual error analysis and to prove the basic stability estimates, it is natural to formulate the problem as a system involving evolution equations for the viscoelastic stress, the displacements, and the velocities. The equations for the viscoelastic stress can, however, be solved analytically in terms of the deviatoric strain velocity, and therefore, the viscoelastic stress can be eliminated from the system, resulting in a system for displacements and velocities.

翻译：本文针对基于空间和时间连续伽辽金法的粘弹性动力学有限元逼近，建立了能量范数和$L^2$范数下的误差估计。证明过程基于离散对偶问题的误差表示公式，以及涉及动能、弹性能和粘弹性能的稳定性估计。为构建对偶误差分析并证明基本稳定性估计，自然地将问题表述为包含粘弹性应力、位移和速度演化方程的系统。然而，粘弹性应力方程可通过偏应变速度解析求解，因此可从系统中消去粘弹性应力，得到仅含位移和速度的方程组。