In this article, we study the semi discrete and fully discrete formulations for a Kirchhoff type quasilinear integro-differential equation involving time-fractional derivative of order $\alpha \in (0,1) $. For the semi discrete formulation of the equation under consideration, we discretize the space domain using a conforming FEM and keep the time variable continuous. We modify the standard Ritz-Volterra projection operator to carry out error analysis for the semi discrete formulation of the considered equation. In general, solutions of the time-fractional partial differential equations (PDEs) have a weak singularity near time $t=0$. Taking this singularity into account, we develop a new linearized fully discrete numerical scheme for the considered equation on a graded mesh in time. We derive a priori bounds on the solution of this fully discrete numerical scheme using a new weighted $H^{1}(\Omega)$ norm. We prove that the developed numerical scheme has an accuracy rate of $O(P^{-1}+N^{-(2-\alpha)})$ in $L^{\infty}(0,T;L^{2}(\Omega))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(\Omega))$, where $P$ and $N$ are degrees of freedom in the space and time directions respectively. The robustness and efficiency of the proposed numerical scheme are demonstrated by some numerical examples.

翻译：本文研究了含$\alpha \in (0,1)$阶时间分数阶导数的Kirchhoff型拟线性积分微分方程的半离散与全离散格式。针对所考虑方程的半离散格式，我们采用协调有限元法对空间域进行离散，并保持时间变量连续。通过修正标准Ritz-Volterra投影算子，完成了所研究方程半离散格式的误差分析。时间分数阶偏微分方程的解通常在$t=0$时刻附近具有弱奇异性。基于该奇异性，我们在时间方向引入渐变网格，为所研究方程构建了一种新的线性化全离散数值格式。通过引入新的加权$H^{1}(\Omega)$范数，推导了该全离散数值格式解的先验界。证明所构建的数值格式在$L^{\infty}(0,T;L^{2}(\Omega))$和$L^{\infty}(0,T;H^{1}_{0}(\Omega))$空间中均具有$O(P^{-1}+N^{-(2-\alpha)})$的收敛阶，其中$P$与$N$分别为空间和时间方向的自由度。数值算例验证了所提数值格式的稳健性与高效性。