Summary: A finite difference scheme for the variable coefficients subdiffusion equations with non-smooth solutions is constructed and analyzed. The spatial derivative is discretized on a uniform mesh, and an \(L1\) approximation is used for the discretization of the fractional time derivative on a possibly graded mesh. The stability of the proposed scheme is given using the discrete energy method. The numerical scheme is \(\mathcal{O} (N^{-\min \{2-\alpha, r\alpha\}})\) accurate in time, where \(\alpha\) \((0 < \alpha < 1)\) is the order of the fractional time derivative, \(r\) is an index of the mesh partition, and it is second order accurate in space. The extension to multi-term time-fractional problems with nonhomogeneous boundary conditions is also discussed, with the stability and error estimate proved both in the discrete \({l^2}\)-norm and the \({l^\infty}\)-norm on the nonuniform temporal mesh. Numerical results are given for both the two-dimensional single and multi-term time-fractional equations.