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：我们针对基于空间和时间的连续伽辽金方法提出的粘弹性动力学有限元近似，证明了能量范数和$L^2$范数下的误差估计。该证明基于一个利用离散对偶问题的误差表示公式，以及一个涉及动能、弹性能和粘弹性能的稳定性估计。为建立对偶误差分析并证明基本的稳定性估计，自然会将问题表述为一个包含粘弹性应力、位移和速度的演化方程组。然而，粘弹性应力方程可依据偏应变速度解析求解，因此可从该方程组中消去粘弹性应力，从而得到一个仅含位移和速度的系统。