稳定化有限元 $\theta$ 格式求解非定常对流占优扩散方程

数学物理学报 ›› 2026, Vol. 46 ›› Issue (3): 1218-1231.

稳定化有限元 $\theta$ 格式求解非定常对流占优扩散方程

孙蓝欣¹, 赖宝伟¹^,², 翁智峰¹^,^*()

¹ 华侨大学数学科学学院, 计算科学福建省高校重点实验室福建泉州 362021
² 北京师范大学文理学院广东珠海 519087

收稿日期:2025-10-26 修回日期:2025-11-27 出版日期:2026-06-26 发布日期:2026-06-16
通讯作者: 翁智峰 E-mail:zfwmath@163.com
基金资助:
国家自然科学基金(11701197);福建省自然科学基金面上项目(2026J001750);福建省自然科学基金面上项目(2025J01167);非线性分析及其应用教育部重点实验室 (华中师范大学) 开放课题(NAA20260RG004);数学与信息网络教育部重点实验室 (北京邮电大学) 开放课题(KF202604)

A Stabilized Finite Element $\theta$ Scheme for Non-Stationary Convection-Dominated Convection Diffusion Problems

Lanxin Sun¹, Baowei Lai¹^,², Zhifeng Weng¹^,^*()

¹ Fujian Province University Key Laboratory of Computation Science, School of Mathematical Sciences, Huaqiao University, Fujian Quanzhou 362021
² Faculty of Arts and Sciences, Beijing Normal University, Guangdong Zhuhai 519087

Received:2025-10-26 Revised:2025-11-27 Online:2026-06-26 Published:2026-06-16
Contact: Zhifeng Weng E-mail:zfwmath@163.com
Supported by:
NSFC(11701197);Natural Science Foundation of Fujian Province(2026J001750);Natural Science Foundation of Fujian Province(2025J01167);Open Research Fund of Key Laboratory of Nonlinear Analysis & Applications (Central China Normal University) Ministry of Education P R China(NAA20260RG004);Open Project of Key Laboratory of Mathematics and Information Networks (Beijing University of Posts and Telecommunications), Ministry of Education China under Grant(KF202604)

摘要/Abstract

摘要：

基于变分多尺度有限元方法, 构造了求解非定常对流占优扩散方程的全离散加权$\theta$格式. 该方法通过采用局部单元上的两局部高斯积分残差代替投影变分多尺度理论框架中的稳定化项. 同时给出了 $L^2$ 范数意义下时空最优误差估计式. 数值算例表明在得到相同的逼近误差时, Crank-Nicolson 格式能够节省计算时间.

关键词: 对流占优扩散方程, 变分多尺度, 稳定化有限元, 两局部高斯积分.

Abstract:

This paper proposes a fully discrete $\theta$ scheme with the variational multiscale finite element method for non-stationary convection-dominated convection diffusion equations. We use an equivalent method based on the residuals of two local Gauss integrations to replace the stabilization term of the variational multiscale method. An optimal error estimate in the space-time $L^2$ norm is also derived. Moreover, numerical results demonstrate that equivalent numerical accuracy can be achieved by the Crank-Nicolson scheme with lower computational cost compared to the Backward Euler scheme.

Key words: convection-dominated convection diffusion equations, variational multiscale method, stabilized finite element method, two local Gauss integrations.

中图分类号:

O241.82

孙蓝欣, 赖宝伟, 翁智峰. 稳定化有限元 $\theta$ 格式求解非定常对流占优扩散方程[J]. 数学物理学报, 2026, 46(3): 1218-1231.

Lanxin Sun, Baowei Lai, Zhifeng Weng. A Stabilized Finite Element $\theta$ Scheme for Non-Stationary Convection-Dominated Convection Diffusion Problems[J]. Acta mathematica scientia,Series A, 2026, 46(3): 1218-1231.

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