In this paper, we are concerned with the numerical solution for the backward fractional Feynman–Kac equation with non-smooth initial data. Here we first provide the regularity estimate of the solution. And then we use the backward Euler and second-order backward difference convolution quadratures to approximate the Riemann–Liouville fractional substantial derivative and get the first- and second-order convergence in time. The finite element method is used to discretize the Laplace operator with the optimal convergence rates. Compared with the previous works for the backward fractional Feynman–Kac equation, the main advantage of the current discretization is that we don’t need the assumption on the regularity of the solution in temporal and spatial directions. Moreover, the error estimates of the time semi-discrete schemes and the fully discrete schemes are also provided. Finally, we perform the numerical experiments to verify the effectiveness of the presented algorithms.

在本文中，我们关注具有非光滑初始数据的向后分数阶Feynman-Kac方程的数值解。在这里，我们首先提供解决方案的规律性估算。然后，我们使用后向Euler和二阶后向差分卷积积分来逼近Riemann-Liouville分数阶实质导数，并及时获得一阶和二阶收敛。使用有限元方法以最佳收敛速度离散化Laplace算子。与先前对后向分数阶Feynman-Kac方程的研究相比，当前离散化的主要优点是我们不需要假设解在时间和空间方向上的规律性。此外，还提供了时间半离散方案和完全离散方案的误差估计。最后，我们进行数值实验以验证所提出算法的有效性。