Abstract This paper is mainly concerned with developing and establishing the reduced-order extrapolated format about the unknown coefficient vectors in numerical solutions to the finite element (FE) method for the hyperbolic type equation. To this end, the functional-form FE format, the existence, stability, and error estimates of the FE solutions, the matrix-form FE format for the hyperbolic type equation are first proposed. Afterwards, a reduced-order extrapolated FE (ROEFE) format is established by means of a proper orthogonal decomposition (POD) technique, and the existence and stability as well as error estimates of the ROEFE solutions are demonstrated by matrix analysis, leading to an elegant theoretical development. Particularly, our work reveals that the ROEFE format possesses the same basis functions and accuracy as the FE method. Finally, some numerical tests are illustrated to computationally confirm the validity and correctness of the ROEFE format.

中文翻译：

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双曲型方程有限元法解未知系数向量的降阶外推技术

摘要 本文主要研究开发和建立双曲型方程有限元方法数值解中未知系数向量的降阶外推格式。为此，首先提出了函数形式的有限元格式，有限元解的存在性、稳定性和误差估计，以及双曲型方程的矩阵形式的有限元格式。然后，通过适当的正交分解（POD）技术建立了降阶外推有限元（ROEFE）格式，并通过矩阵分析证明了ROEFE解的存在性和稳定性以及误差估计，从而得到了一个优雅的理论发展。特别是，我们的工作表明 ROEFE 格式具有与 FE 方法相同的基函数和精度。最后，