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 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 first-order-in-time 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.

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粘弹性动力学有限元近似的误差估计：广义麦克斯韦模型

我们证明了基于空间和时间连续伽辽金的粘弹性动力学有限元近似的误差估计，包括能量范数和范数。该证明基于使用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。为了建立对偶误差分析并证明基本的稳定性估计，很自然地将问题表述为涉及粘弹性应力、位移和速度的演化方程的一阶时间系统。然而，粘弹性应力方程可以根据偏应变速度进行解析求解，因此，可以从系统中消除粘弹性应力，从而产生位移和速度系统。