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Application of spectral element method for solving Sobolev equations with error estimation
Applied Numerical Mathematics ( IF 2.2 ) Pub Date : 2020-12-01 , DOI: 10.1016/j.apnum.2020.08.010
Mehdi Dehghan , Nasim Shafieeabyaneh , Mostafa Abbaszadeh

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:

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。
更新日期:2020-12-01
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