We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end.

中文翻译：

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涉及分数阶积分微分本构律的动态粘弹性问题有限元近似的先验误差分析

我们考虑由幂律型应力松弛函数建模的分数阶粘弹性问题。此粘弹性问题是第二类 Volterra 积分方程，具有弱奇异核，其中卷积积分对应于分数阶微分/积分。我们使用空间有限元方法和时间上的有限差分方案。由于弱奇异性，时间上的分数阶积分近似通过线性插值进行管理，以便我们可以制定一个完全离散的问题。在本文中，我们提出了稳定性界限和先验误差估计。此外，我们在最后进行了具有不同精确解规律的数值实验。