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Superconvergence in H 1 -norm of a difference finite element method for the heat equation in a 3D spatial domain with almost-uniform mesh
Numerical Algorithms ( IF 1.7 ) Pub Date : 2020-03-02 , DOI: 10.1007/s11075-020-00892-y
Xinlong Feng , Ruijian He , Zhangxin Chen

In this paper, we propose a novel difference finite element (DFE) method based on the P1-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of this paper is to use the second-order backward difference formula (BDF) combining DFE method to overcome the computational complexity of conventional finite element (FE) method for the high-dimensional parabolic problem. First, we design a fully discrete difference FE solution \({u^{n}_{h}}\) by the second-order backward difference formula in the temporal t-direction, the center difference scheme in the spatial z-direction, and the P1-element on a almost-uniform mesh Jh in the spatial (x, y)-direction. Next, the H1-stability of \({u_{h}^{n}}\) and the second-order H1-convergence of the interpolation post-processing function on \({u_{h}^{n}}\) with respect to u(tn) are provided. Finally, numerical tests are presented to show the second-order H1-convergence results of the proposed DFE method for the heat equation in a 3D spatial domain.



中文翻译:

几乎均匀网格的3D空间域中热方程的差分有限元方法在H 1范数中的超收敛

在本文中,我们针对3D有界域上的3D热方程提出了一种基于P 1元素的新型差分有限元(DFE)方法。本文的新颖思想之一是结合DFE方法使用二阶后向差分公式(BDF)来克服传统有限元(FE)方法在高维抛物线问题上的计算复杂性。首先,我们通过时间t方向上的二阶后向差分公式,空间z方向上的中心差分方案设计完全离散的差分FE解\({u ^ {n} _ {h}} \),以及几乎均匀的网格J h上的P 1元素在空间(xy)方向上。接着,ħ 1的-稳定性\({U_ {H} ^ {N}} \)和二阶ħ 1的后处理内插函数的上-convergence \({U_ {H} ^ {N} } \)关于ut n)被提供。最后,数值试验表明了所提出的DFE方法在3D空间域中的热方程的二阶H 1收敛结果。

更新日期:2020-03-02
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