  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.
