针对含有变系数的二维分布阶扩散方程,利用高斯积分对分布阶算子$D_t^\omega u$进行逼近,则原问题变为一个多项时间分数阶微分方程.在空间方向上主要采用非协调的$EQ_1^{\rm rot}$元和零阶Raviart-Thomas (R-T)元,在时间方向上运用修正的L1格式,建立了全离散逼近格式,进一步证明全离散格式的稳定性.利用单元的性质以及插值算子$\Pi_h$、$I_h$和投影算子$R_h$的性质,分别得到$H^1$模下变量$u$和$L^2$模下中间变量$\overrightarrow{p}=\hbar (X)\nabla u$的超逼近结果.最后,根据插值算子$I_{2h}$和$\Pi_{2h}$的相关性质,得到了整体超收敛结果.
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 $\Pi_{2h}$.
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