对流扩散反应方程的局部投影稳定化连续时空有限元方法

doi:10.12286/jssx.j2019-0641

计算数学 ›› 2021, Vol. 43 ›› Issue (3): 367-387.DOI: 10.12286/jssx.j2019-0641

对流扩散反应方程的局部投影稳定化连续时空有限元方法

董自明^1,2, 李宏¹, 赵智慧¹, 唐斯琴¹

1. 内蒙古大学数学科学学院, 呼和浩特 010021;
2. 包头师范学院数学科学学院, 包头 014030

收稿日期:2019-11-06 出版日期:2021-08-15 发布日期:2021-08-20
通讯作者: 李宏,E-mail:malhong@imu.edu.cn.
基金资助:
由国家自然科学基金（11761053），内蒙古自然科学基金（2021MS01018，2019BS01010），内蒙古自治区草原英才，内蒙古自治区青年科技英才-领军人才项目（NJYT-17-A07）资助.

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董自明, 李宏, 赵智慧, 唐斯琴. 对流扩散反应方程的局部投影稳定化连续时空有限元方法[J]. 计算数学, 2021, 43(3): 367-387.

Dong Ziming, Li Hong, Zhao Zhihui, Tang Siqin. LOCAL PROJECTION STABILIZATION CONTINUOUS SPACE-TIME FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION-REACTION EQUATIONS[J]. Mathematica Numerica Sinica, 2021, 43(3): 367-387.

LOCAL PROJECTION STABILIZATION CONTINUOUS SPACE-TIME FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION-REACTION EQUATIONS

Dong Ziming^1,2, Li Hong¹, Zhao Zhihui¹, Tang Siqin¹

1. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China;
2. Faculty of Mathematics, Baotou Teachers'College, Baotou 014030, China

Received:2019-11-06 Online:2021-08-15 Published:2021-08-20

本文将局部投影稳定化（LPS）方法和连续时空有限元方法相结合研究对流扩散反应方程，给出稳定化连续时空有限元离散格式.与传统的时空有限元研究思路不同，时间方向利用Lagrange插值多项式，解耦时间和空间变量，降低时空有限元解的维数，具有减少计算量和简化理论分析的优点.通过引入Legendre多项式给出了有限元解的稳定性分析，进一步引进Lobatto多项式证明了有限元解的全局L^∞（L²）和局部L²（J_n；LPS）范数误差估计.最后给出数值算例验证理论分析的正确性，以及稳定化格式的可行性和有效性.

关键词： 对流扩散反应方程, LPS方法, 连续时空有限元方法, 误差估计

In this paper, local projection stabilization method and continuous space-time finite element method are combined to study convection-diffusion-reaction equations. The discrete form of stabilized continuous space-time Galerkin method is constructed. The ideas discussed here are different from the traditional space-time finite element method. The approaches presented here have the advantages of reducing calculation and simplifying theoretical analysis with the techniques of Lagrange interpolation polynomials in time direction, which not only can decouple time and space variables but also reduce the dimensions of the spacetime finite element solution. The stability analysis of finite element solution is obtained by Legendre polynomials. Moreover, the error estimates in global L^∞(L²)-norm and local L²(^J_n; LPS)-norm are proved with Lobatto polynomials. Finally, numerical examples are given to verify correctness of the theoretical analysis and feasibility and validity of the stabilization scheme.

Key words: convection-diffusion-reaction equations, local projection stabilization method, continuous Galerkin method, error estimates

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摘要

对流扩散反应方程的局部投影稳定化连续时空有限元方法

LOCAL PROJECTION STABILIZATION CONTINUOUS SPACE-TIME FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION-REACTION EQUATIONS

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