This work constructs and analyzes a nonlocal evolution equation with a weakly singular kernel in three-dimensional space. In the temporal direction, the Crank-Nicolson (CN) method and product-integration (PI) rule are employed, from which the non-uniform meshes are used to eliminate the singular behaviour of the exact solution at $t=0$. Then, a fully discrete scheme is obtained by the spatial discretization based on the finite difference method. Simultaneously, an alternating direction implicit (ADI) algorithm is designed to reduce the computational cost. The stability in $L^2$ norm and convergence are derived via the energy method, in which the convergence orders are $\mathcal{O}(k^2+h^2)$ with certain suitable assumptions, where k and h are corresponding space-time step sizes, respectively. Numerical results confirm the theoretical analysis.

这项工作构建并分析了一个在三维空间中具有弱奇异核的非局部演化方程。在时间方向上，采用 Crank-Nicolson (CN) 方法和乘积积分 (PI) 规则，从中使用非均匀网格消除精确解在$吨=0$. 然后，通过基于有限差分法的空间离散化得到一个完全离散的方案。同时，设计了交替方向隐式（ADI）算法以降低计算成本。中的稳定性${升}^2$ 范数和收敛是通过能量方法导出的，其中收敛阶数是 $\mathcal{哦}({克}^2+H^2)$具有某些合适的假设，其中k和h分别是相应的时空步长。数值结果证实了理论分析。