This study investigated the effects of 5E instructional model on the teaching processes of novice teachers. First, we conducted a teaching design training project based on the 5E model for 40 novice teachers, and compared pre-texts of the teachers' teaching process from before the training with post-texts obtained immediately following the training to determine whether the model can promote the teaching design process of novice teachers. In order to explore how the 5E model influenced the novice teachers' teaching processes, we chose three teachers for an additional three stages of training, then compared the texts resulting from the different stages and interviewed each teacher. Finally, we found that the 5E model had a significant effect on the improvement and further development of the teaching processes among the novice teachers. The model influenced the teachers' teaching process through each of the sub-phases, and different sub-phases resulted in different improvements. Each novice teacher also showed different improvements, with the specific improvements also being affected by teachers' personal beliefs.

Time fractional PDEs have been used in many applications for modeling and simulations. Many of these applications are multiscale and contain high contrast variations in the media properties. It requires very small time step size to perform detailed computations. On the other hand, in the presence of small spatial grids, very small time step size is required for explicit methods. Explicit methods have many advantages as we discuss in the paper. In this paper, we propose a partial explicit method for time fractional PDEs. The approach solves the forward problem on a coarse computational grid, which is much larger than spatial heterogeneities, and requires only a few degrees of freedom to be treated implicitly. Via the construction of appropriate spaces and careful stability analysis, we can show that the time step can be chosen not to depend on the contrast or scale as the coarse mesh size. Thus, one can use larger time step size in an explicit approach. We present stability theory for our proposed method and our numerical results confirm the stability findings and demonstrate the performance of the approach.

We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data a(x) ∈ L 2 (D) in a bounded domain D ⊂ R d with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient κ (x) is smooth and periodic with the period > 0 being sufficiently small. We derive that its first order approximation has a convergence rate of O( 1/2 ) when the dimension d ≤ 2 and O( 1/6 ) when d = 3. Several numerical tests are presented to show the performance of the first order approximation.

In this paper, we consider the incompressible Stokes flow problem in a perforated domain and employ the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to solve this problem. The proposed method provides a flexible and systematical approach to construct crucial divergence-free multiscale basis functions for approximating the displacement field. These basis functions are constructed by solving a class of local energy minimization problems over the eigenspaces that contain local information on the heterogeneities. These multiscale basis functions are shown to have the property of exponential decay outside the corresponding local oversampling regions. By adapting the technique of oversampling, the spectral convergence of the method with error bounds related to the coarse mesh size is proved.

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