This study is mainly concerned with the reduced-order extrapolating technique about the unknown solution coefficient vectors in the Crank-Nicolson finite element (CNFE) method for the parabolic type partial differential equation (PDE). For this purpose, the CNFE method and the existence, stability, and error estimates about the CNFE solutions for the parabolic type PDE are first derived. Next, a reduced-order extrapolating CNFE (ROECNFE) model in matrix-form is established with a proper orthogonal decomposition (POD) method, and the existence, stability, and error estimates of the ROECNFE solutions are proved by matrix theory, resulting in an graceful theoretical development. Specially, our study exposes that the ROECNFE method has the same basis functions and the same accuracy as the CNFE method. Lastly, some numeric tests are shown to computationally verify the validity and correctness about the ROECNFE method.

中文翻译：

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抛物线型方程的Crank-Nicolson有限元解系数向量的降阶外推法

该研究主要涉及抛物型偏微分方程（PDE）的Crank-Nicolson有限元（CNFE）方法中有关未知解系数向量的降阶外推技术。为此，首先导出CNFE方法以及关于抛物线型PDE的CNFE解的存在性，稳定性和误差估计。接下来，使用适当的正交分解（POD）方法建立矩阵形式的降阶外推CNFE（ROECNFE）模型，并通过矩阵理论证明ROECNFE解的存在性，稳定性和误差估计，从而得出优美的理论发展。特别地，我们的研究表明ROECNFE方法具有与CNFE方法相同的基本功能和相同的精度。最后，