Two numerical methods with graded temporal grids are analyzed for fractional evolution equations. One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order \(0<\alpha <1\), and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order \(1<\alpha <2\). By a new duality technique, pointwise-in-time error estimates of first-order and \( (3-\alpha ) \)-order temporal accuracies are respectively derived for DG and PG, under reasonable regularity assumptions on the initial value. Numerical experiments are performed to verify the theoretical results.

中文翻译：

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分数阶演化方程的两个带梯度时间网格的Galerkin离散化的数值分析

分析了分数阶演化方程的两种带有渐变时间网格的数值方法。一个是分数阶\（0 <\ alpha <1 \）的低阶不连续Galerkin（DG）离散，另一个是分数阶数的低阶Petrov Galerkin（PG）离散\（1 <\ alpha <2 \）。通过一种新的对偶技术，在合理的初始值假设的前提下，分别为DG和PG导出了一阶和\（（3- \ alpha）\）阶时间精度的时间点实时误差估计。进行数值实验以验证理论结果。