Two time-stepping schemes for sub-diffusion equations with singular source terms

The singular source terms in sub-diffusion equations may lead to the unboundedness of the solutions, which will bring a severe reduction of convergence order of the existing numerical schemes. In this work, we propose two 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 under a weak regularity of both the source term and initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess an improved order of accuracy. Further, second-order accurate finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.