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The solution of time fractional partial differential equations in generalexhibit a weak singularity near the initial time. In this article we propose amethod for solving time fractional diffusion equation with nonlocal diffusionterm. The proposed method comprises L1 scheme on graded mesh, finite elementmethod and Newton's method. We discuss the well-posedness of the weakformulation at discrete level and derive \emph{a priori} error estimates forfully-discrete formulation in $L^2(\Omega)$ and $H^1(\Omega)$ norms. Finally,some numerical experiments are conducted to validate the theoretical findings.

L1 scheme on graded mesh for subdiffusion equation with nonlocal diffusion term