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In this work, a time-fractional diffusion problem with a time-space dependent diffusivity is considered. The solution of such a problem has a weak singularity at the initial time t = 0 $t=0$ . Based on the L1 scheme in time on a graded mesh and the conforming finite element method in space on a uniform mesh, the fully discrete L1 conforming finite element method (L1 FEM) of a time-fractional diffusion problem is investigated. The error analysis is based on a nonstandard discrete Gronwalldoi:10.1186/s13662-020-02976-4 doaj:ffd8c9c1f56548728e2345ff59616ce5 fatcat:pjhnch5wufgg3gynssrmorhog4