The main aim of this paper is to propose a 2-step backward differential formula (BDF2) fully discrete scheme with the bilinear Q11 finite element method (FEM) for the nonlinear reaction–diffusion equation. By use of the combination technique of the element’s interpolation and Ritz projection, and through the interpolation post-processing approach, the superclose and global superconvergence estimates with order O(h2+τ2) in H1-norm are deduced rigorously. Furthermore, with the help of the asymptotic error expansion of the Q11 element, a new suitable fully discrete scheme is developed, and the extrapolation result of order O(h3+τ2) in H1-norm is derived, which is one order higher than that of the above traditional superconvergence estimate with respect to h. Here h is the mesh size and τ is the time step. Finally, some numerical results are provided to verify the theoretical analysis. It seems that the extrapolation of the fully discrete finite element scheme has never been seen in the previous studies.

中文翻译：

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非线性反应-扩散方程的 BDF2 全离散方案的超收敛分析和外推


本文的主要目的是提出一种 2 步后退微分公式 （BDF2） 全离散方案，其中双线性 Q11 有限元法 （FEM） 用于非线性反应-扩散方程。通过使用单元插值和 Ritz 投影的组合技术，以及通过插值后处理方法，严格推导了 H1 范数中阶数为 O（h2+τ2） 的超接近和全局超收敛估计。此外，借助Q11元件的渐近误差展开，开发了一种新的合适的全离散方案，推导出了H1范数中O（h3+τ2）阶的外推结果，比上述传统超收敛估计对h的外推结果高了一个数量级。其中 h 是网格大小，τ 是时间步长。最后，给出了一些数值结果，验证了理论分析的有效性。在以前的研究中，似乎从未见过完全离散有限元方案的外推。