A novel energy dissipation Crank-Nicolson (C-N) fully discrete scheme is established by low order nonconforming EQ1rot element for solving the Sobolev equations with high order Burgers' type nonlinearity. Firstly, the boundedness of the discrete solution in the broken H1-norm is achieved directly by the energy dissipation property without using the known time-space splitting technique in the existing literatures, and its well-posedness is demonstrated by the Brouwer fixed point theorem. Secondly, by utilizing the special characters of nonconforming EQ1rot element, the unconditional superclose result of order O(h2+τ2) in the broken H1-norm is gained strictly with no restrictions between the spatial partition parameter h and the time step τ. Moreover, the corresponding global superconvergent error estimate of order O(h2+τ2) is proved by applying an interpolation post-processing approach. Thirdly, an application to some different finite elements and nonlinear PDEs is discussed, which shows that the proposed scheme and the analysis presented herein can be considered as a general framework to cope with. Lastly, the theoretical results are validated by four numerical examples.

中文翻译：

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高阶 Burgers 型非线性 Sobolev 方程的新型能量耗散非整合型 Crank-Nicolson FEM 的无条件超收敛分析


利用低阶非一致性 EQ1rot 单元建立了一种新的能量耗散 Crank-Nicolson （C-N） 全离散方案，用于求解具有高阶 Burgers 型非线性的 Sobolev 方程。首先，在不使用现有文献中已知的时空分裂技术的情况下，通过能量耗散特性直接实现了离散解的有界性，并通过 Brouwer 不动点定理证明了其适定性。其次，利用不合格 EQ1rot 元素的特殊字符，在打破的 H1 范数中严格获得阶数 O（h2+τ2） 的无条件超闭合结果，空间分区参数 h 和时间步长 τ 之间没有限制。此外，通过应用插值后处理方法证明了相应的 O（h2+τ2） 阶全局超收敛误差估计。然后，讨论了在一些不同的有限元和非线性偏微分方程上的应用，这表明所提出的方案和本文提出的分析可以被视为一个通用的框架来应对。最后，通过4个数值算例验证了理论结果。