Abstract This paper is dedicated to numerically solving the Sobolev equations that have several applications in physics and mechanical engineering. First, the temporal derivative is discretized by the Crank-Nicolson finite difference technique to obtain a semi-discrete scheme in the temporal direction. Afterward, the stability and convergence analysis of the time semi-discrete scheme are proven by applying the energy method. It also implies that the convergence order in the temporal direction is O ( d t 2 ) . Second, a fully discrete formula has been acquired by discretizing the spatial derivatives via Legendre spectral element method (LSEM). This method applies the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points. Moreover, an error estimation is given for the obtained fully discrete scheme. Eventually, the two-dimensional Sobolev equations are solved by using the proposed procedure. The accuracy and efficiency of the mentioned procedure are demonstrated by several numerical examples.

中文翻译：

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谱元法在求解带误差估计的Sobolev方程中的应用

摘要 本文致力于数值求解在物理和机械工程中具有多种应用的 Sobolev 方程。首先，通过Crank-Nicolson有限差分技术对时间导数进行离散化，得到时间方向上的半离散方案。然后，应用能量法证明了时间半离散方案的稳定性和收敛性分析。这也意味着时间方向上的收敛顺序是 O (dt 2 ) 。其次，通过勒让德谱元法 (LSEM) 对空间导数进行离散化，获得了完全离散的公式。此方法应用基于 Gauss-Legendre-Lobatto (GLL) 点的拉格朗日多项式。此外，对获得的完全离散方案给出了误差估计。最终，二维 Sobolev 方程是通过使用所提出的程序求解的。几个数值例子证明了上述程序的准确性和效率。