Two Time-Stepping Schemes for Sub-Diffusion Equations with Singular Source Terms

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