二维时间分数阶扩散方程的Hermite型矩形元的超收敛分析

基于经典的L1逼近,针对二维时间分数阶扩散方程给出Hermite型矩形元的全离散格式.首先,证明其逼近格式的无条件稳定性.其次,基于Hermite型矩形元的积分恒等式结果,建立插值与Ritz投影之间在H 1模意义下的超收敛估计.进而,通过利用插值与投影的关系及巧妙地处理分数阶导数,得到单独利用插值或Ritz投影所无法得到的超逼近及超收敛结果.最后,借助于插值后处理技术导出了整体超收敛结果.

Superconvergence Analysis of Hermite-Type Rectangular Element Method for Two-Dimensional Time Fractional Diffusion Equations

Based on the classical L1 approximation scheme, a Hermite-type rectangular element method is proposed for two-dimensional time fractional diffusion equations under the fully-discrete scheme. Firstly, unconditional stability analysis of the approcimate scheme is provided. Secondly, by use of the integral indentity result of Hermite-type rectangular element, superconvergence estimate in H1-norm is established between the interpolation and Ritz projection. Moreover, combining with the relationship between the interpolation operator and Ritz projection, and by dealing with fractional derivatives skillfully, superclose and superconvergence results are obtained, which cann’t be deduced by interpolation or Ritz projection alone. Finally, the global superconvergence property is derived by the technique of the postprocessing operator.