A three step backward differential formula scheme is proposed for nonlinear reaction–diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size and the subdivision parameter . Temporal error estimate in H²-norm is derived, which leads to the boundedness of the solutions of the time-discrete equations. Unconditional spatial error estimate in L²-norm is deduced which help bound the numerical solutions in L^∞-norm. Superconvergent property of in H¹-norm with order is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.

中文翻译：

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非线性反应-扩散方程的三步后向微分公式-有限元法能量稳定方案的超收敛性分析

提出了非线性反应-扩散方程的三步后向微分公式方案，并无条件地用Galerkin有限元法研究了超收敛结果。用人工项验证了所构建方案的能量稳定性。利用分割技术消除时间步长与细分参数的比例关系。推导了H ²范数下的时间误差估计，这导致了时间离散方程解的有界性。推导了L ² -范数中的无条件空间误差估计，这有助于限制L ^∞ -范数中的数值解。in H的超收敛性质通过无条件地取误差方程的两个时间水平之间的差来获得有阶的^1-范数。通过以上结果推导出全局超收敛性。两个数值例子表明了理论分析的有效性。