二阶双曲方程的全离散格式下的混合元超收敛分析

doi:10.12286/jssx.j2021-0795

计算数学 ›› 2023, Vol. 45 ›› Issue (1): 8-21.DOI: 10.12286/jssx.j2021-0795

二阶双曲方程的全离散格式下的混合元超收敛分析

杨怀君

郑州航空工业管理学院, 数学学院, 郑州 450046

收稿日期:2021-03-27 出版日期:2023-02-14 发布日期:2023-02-13
基金资助:
国家自然科学基金(12101568)和郑州航空工业管理学院博士启动基金(63020390)资助.

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杨怀君. 二阶双曲方程的全离散格式下的混合元超收敛分析[J]. 计算数学, 2023, 45(1): 8-21.

Yang Huaijun. SUPERCONVERGENCE ANALYSIS OF MIXED FINITE ELEMENT METHOD FOR SECOND-ORDER HYPERBOLIC EQUATION WITH FULLY-DISCRETE SCHEME[J]. Mathematica Numerica Sinica, 2023, 45(1): 8-21.

SUPERCONVERGENCE ANALYSIS OF MIXED FINITE ELEMENT METHOD FOR SECOND-ORDER HYPERBOLIC EQUATION WITH FULLY-DISCRETE SCHEME

Yang Huaijun

School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450046, China

Received:2021-03-27 Online:2023-02-14 Published:2023-02-13
Supported by:
The project was supported by the National Key Research and Development Program of China (2019YFC1905301);National Natural Science Foundation of China (22078115,21776108,21690083,22008078).

通过在空间方向上使用双线性元和最低阶的 Nedeléc 元 (即Q₁₁ + Q₀₁ × Q₁₀)以及在时间方向上使用二阶精度的数值逼近格式, 得到了在矩形网格上二阶双曲方程全离散混合元格式下的对原始变量的L^∞(H¹) 和流量的L^∞((L²)²)的超逼近和超收敛的误差结果. 在分析过程中, 巧妙地使用了上述混合单元对在矩形网格上的特有的高精度积分恒等式和精确解的投影和插值之间的在H¹范数意义下的超逼近的估计. 最后, 给出一些数值结果来验证理论分析的正确性.

关键词： 二阶双曲方程, 混合元全离散, 超逼近和超收敛

Based on the bilinear element and the lowest order Nedeléc element (i.e. Q₁₁+Q₀₁×Q₁₀) used in spatial direction and a second order accurate scheme applied in temporal direction, the superclose and superconvergence error estimates for original variable in L^∞(H¹) and for flux variables in L^∞((L²)²) are obtained for second order hyperbolic equation on the rectangular mesh. The analysis relies on the integral identity of the element pair and the technique of combining the interpolation and projection operators as well as the superclose estimate in H¹-norm between them. Finally, some numerical results are provided to confirm the theoretical analysis.

Key words: Hyperbolic equation, fully-discrete scheme, superclose and superconvergence error estimates

MR(2010)主题分类:

65N15
65N30

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二阶双曲方程的全离散格式下的混合元超收敛分析

SUPERCONVERGENCE ANALYSIS OF MIXED FINITE ELEMENT METHOD FOR SECOND-ORDER HYPERBOLIC EQUATION WITH FULLY-DISCRETE SCHEME

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