粘弹性波动方程的H$^1$-Galerkin时空混合有限元分裂格式

doi:10.12286/jssx.j2021-0892

计算数学 ›› 2023, Vol. 45 ›› Issue (2): 177-196.DOI: 10.12286/jssx.j2021-0892

粘弹性波动方程的H$^1$-Galerkin时空混合有限元分裂格式

王嘉华, 李宏

内蒙古大学数学科学学院, 呼和浩特, 010021

收稿日期:2021-12-14 出版日期:2023-05-14 发布日期:2023-05-13
通讯作者: 李宏, E-mail:smslh@imu.edu.cn
基金资助:
国家自然科学基金(12161063), 内蒙古自然科学基金(2021MS01018)和内蒙古自治区高等学校创新团队发展计划(NMGIRT2207)资助.

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王嘉华, 李宏. 粘弹性波动方程的H$^1$-Galerkin时空混合有限元分裂格式[J]. 计算数学, 2023, 45(2): 177-196.

Wang Jiahua, Li Hong. H¹-GAlERKIN SPACE TIME MIXED FINITE ELEMENT SPLITTING SCHEME FOR ONE-DIMENSIONAL VISCOELASTIC WAVE EQUATION[J]. Mathematica Numerica Sinica, 2023, 45(2): 177-196.

H¹-GAlERKIN SPACE TIME MIXED FINITE ELEMENT SPLITTING SCHEME FOR ONE-DIMENSIONAL VISCOELASTIC WAVE EQUATION

Wang Jiahua, Li Hong

School of Mathematical Science, Inner Mongolia University, Hohhot 010021, China

Received:2021-12-14 Online:2023-05-14 Published:2023-05-13

构造一维粘弹性波动方程的H$^1$-Galerkin时空有限元分裂格式. 这种新的分裂格式在时空两个方向同时利用有限元离散, 具有H$^1$-Galerkin 混合有限元方法和时空有限元方法的优点, 如在不受LBB 相容性条件限制的同时能够高精度逼近流体的压力和达西速度, 有限元空间可以利用不同次数的多项式空间, 能同时得到时间和空间两个变量的形式高阶精度等. 通过构造时空投影算子并讨论其相关逼近性质, 证明了解的存在唯一性和稳定性, 给出混合时空有限元解的误差估计, 给出数值算例验证了理论推导结果的合理性和算法的有效性,并和传统H$^1$-Galerkin方法做比较,得到了更小的误差和超收敛阶.

关键词： 粘弹性波动方程, H$^1$-Galerkin时空混合有限元方法, 分裂格式, 存在唯一性和稳定性, 误差估计, 数值算例

An H¹-Galerkin space time mixed finite element splitting scheme is constructed for one dimensional viscoelastic wave equations.The finite element discrete are used in both space and time direction in the new splitting scheme, and the approximate method inherits the advantages of H¹-Galerkin mixed finite element method and space time finite element method.For example, it can approximate the fluid pressure and Darcy Velocity with high accuracy without limited LBB compatibility condition. The polynomial space of different degree can be used as the finite element space. The formal high order accuracy of the time and space variables can be obtained simultaneously. The existence and uniqueness, error estimation of the space time mixed finite element solution are proved by introducing the space time projections and discussing the properties of them. Numerical examples are given to verify the rationality of the theoretical results and the efficiency of the algorithm. In the meantime, comparing between the constructed space time scheme and traditional H¹-Galerkin mixed finite element method, better error and super convergence order are obtained for the constructed space time scheme in this paper.

Key words: viscoelastic wave equation, H¹-Galerkin space time mixed finite element method, splitting scheme, stability and existence as well as uniqueness, error estimates, numerical examples

MR(2010)主题分类:

65N30

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粘弹性波动方程的H$^1$-Galerkin时空混合有限元分裂格式

H1-GAlERKIN SPACE TIME MIXED FINITE ELEMENT SPLITTING SCHEME FOR ONE-DIMENSIONAL VISCOELASTIC WAVE EQUATION

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H¹-GAlERKIN SPACE TIME MIXED FINITE ELEMENT SPLITTING SCHEME FOR ONE-DIMENSIONAL VISCOELASTIC WAVE EQUATION