  Singular source terms in sub-diffusion equations may lead to the
unboundedness of solutions, which will bring a severe reduction of convergence
order of existing time-stepping schemes. In this work, we propose two efficient
time-stepping schemes for solving sub-diffusion equations with a class of
source terms mildly singular in time. One discretization is based on the
Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate
with respect to time is rigorously established for singular source terms and
nonsmooth initial data. The other scheme derived from the second-order backward
differentiation formula (BDF) is proved to possess second-order accuracy in
time. Further, piecewise linear finite element and lumped mass finite element
discretizations in space are applied and analyzed rigorously. Numerical
investigations confirm our theoretical results.
