Abstract: For the two-dimensional distributed-order time fractional diffusion equation with a variable coefficient in this paper, a Gauss integral approximates the distributed-order operator $D^\omega_t u$ and original problem, which is transformed into a multi-term time fractional differential equation. The nonconforming $EQ_1^{\rm rot}$ and zero-order Raviart-Thomas (R-T) elements are employed in a spatial direction, the modified L1 scheme is applied in a temporal direction, the fully discrete scheme of the equation is established, and the stability of the fully discrete scheme is then demonstrated. Using the interpolation operator $\Pi_h$, $I_h$ and projection operator $R_h$, of the elements, the superclose results of the variable $u$ in $H^1$-norm and intermediate variable $\overrightarrow{p}=\hbar (X)\nabla u$ in $L^2$-norm are obtained, respectively. Finally, the global superconvergence results are derived by using the related properties of the interpolation operators $I_{2h}$ and $\Pi_{2h}$.

Key words: diffusion equation, nonconforming, stability, superclose, superconvergence