非线性sine-Gordon方程的连续时空混合有限元方法

王媋瑗; 李宏; 何斯日古楞

doi:10.21656/1000-0887.440293

非线性sine-Gordon方程的连续时空混合有限元方法

doi: 10.21656/1000-0887.440293

1.
内蒙古大学数学科学学院, 呼和浩特 010021
2.
呼和浩特民族学院数学科学学院, 呼和浩特 010051

基金项目:

国家自然科学基金 12161063

国家自然科学基金 12161034

内蒙古自然科学基金 2021MS01018

内蒙古自治区高等学校创新团队发展计划 NMGIRT2207

详细信息

作者简介:
王媋瑗(1996—), 女, 硕士生(E-mail：cywang0524@163.com)

通讯作者:
李宏(1973—), 女, 教授, 博士生导师(通讯作者. E-mail: smslh@imu.edu.cn)

中图分类号: O357.4⁺1
计量
- 文章访问数: 322
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- PDF下载量: 50
- 被引次数: 0
出版历程
- 收稿日期: 2023-09-27
- 修回日期: 2023-11-28
- 刊出日期: 2024-04-01

A Continuous Space-Time Mixed Finite Element Method for Sine-Gordon Equations

1.
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, P.R.China
2.
School of Mathematical Sciences, Hohhot Minzu College, Hohhot 010051, P.R.China

摘要

摘要: 该文将混合有限元方法和连续时空有限元方法相结合, 构造了sine-Gordon方程的连续时空混合有限元离散格式, 引入独立变量p=u_t来求解, 并将时间变量和空间变量都用有限元方法处理. 此格式可以将方程降阶, 降低有限元空间的光滑性要求, 同时在时间和空间两个方向都能发挥有限元方法的优势, 获得时空高精度的数值解. 理论分析中严格证明了数值解的稳定性, 给出了u和p的误差估计. 最后通过数值算例的结果展示了格式的有效性和可行性.
- sine-Gordon方程 /
- 连续时空混合有限元 /
- 稳定性 /
- 误差估计
Abstract: The mixed finite element method was combined with the continuous space-time finite element method to construct a continuous space-time mixed finite element scheme for sine-Gordon equations, through the introduction of independent variable p=u_t to solve the equations. This scheme uses the finite element method to treat both time and space variables. The space-time mixed finite element scheme can reduce the order of the equation and lower the smoothness requirements on the finite element space. The advantages of the finite element method was utilized in both the time and the space directions, thereby to obtain high-precision space-time numerical solutions. The stability of numerical solutions was strictly proven in the theoretical analysis, and error estimates for u and p were provided. Finally, the effectiveness and feasibility of the proposed method were demonstrated through numerical examples.
- sine-Gordon equation /
- continuous space-time mixed finite element method /
- stability /
- error estimate

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图 1 线性基函数, u和u^hk的对比

Figure 1. Comparison between u and u^hk with linear basis function

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图 2 线性基函数, p和p^hk的对比

Figure 2. Comparison between p and p^hk with linear basis function

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图 3 二次基函数, u和u^hk的对比(t=T)

Figure 3. Comparison between u and u^hk with the quadratic basis function(t=T)

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图 4 二次基函数, p和p^hk的对比(t=T)

Figure 4. Comparison between p and p^hk with the quadratic basis function(t=T)

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图 5 线性基函数, t=T的收敛阶对比

Figure 5. Comparison of convergence rates at time t=T for the linear basis function

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图 6 二次基函数, t=T的收敛阶对比

Figure 6. Comparison of convergence rates at time t=T for the quadratic basis function

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表 1 用线性基函数和k=1/200时, 关于空间方向的误差和收敛阶

Table 1. Error and convergence rates in the space direction with the linear basis function and k=1/200

h	‖p-p^hk‖_a	rate log₂(k)	‖u-u^hk‖_a	rate log₂(k)
1/4	4.739 6E-3		1.781 9E-3
1/8	1.330 1E-3	1.833 3	4.985 6E-4	1.837 6
1/16	3.409 2E-4	1.964 0	1.278 7E-4	1.963 1
1/32	8.578 9E-5	1.990 6	3.217 3E-5	1.990 8

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表 2 用线性基函数和k=1/200时, 关于时间方向的误差和收敛阶

Table 2. Error and convergence rates in the time direction with the linear basis function and k=1/200

h	‖p-p^hk‖_a	rate log₂(k)	‖u-u^hk‖_a	rate log₂(k)
1/4	8.072 5E-4		3.590 1E-4
1/8	2.537 4E-4	1.669 7	9.153 6E-5	1.971 6
1/16	6.484 0E-5	1.968 4	2.293 7E-5	1.996 7
1/32	1.664 3E-5	1.962 0	5.793 9E-6	1.985 0

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表 3 线性基函数, t=T时刻的误差和收敛阶

Table 3. Error and convergence rates at time t=T for the linear basis function

k(k=10h)	1/4	1/8	1/16	1/32
‖u-u_CN‖_b	4.035 9E-3	1.128 9E-3	2.892 0E-4	7.272 5E-5
rate log₂(k)		1.837 9	1.964 8	1.991 5
‖u-u_ST‖_b	1.782 0E-3	4.985 7E-4	1.278 7E-4	3.217 2E-5
rate log₂(k)		1.837 6	1.963 1	1.990 8
‖p-p_CN‖_b	8.669 2E-3	2.375 9E-3	6.010 2E-4	1.505 1E-4
rate log₂(k)		1.867 4	1.982 9	1.997 5
‖p-p_ST‖_b	8.497 0E-3	2.365 9E-3	6.001 4E-4	1.504 6E-4
rate log₂(k)		1.826 4	1.968 9	1.995 7

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表 4 二次基函数, t=T时刻的误差和收敛阶

Table 4. Error and convergence rates at time t=T for the quadratic basis function

k(k=10h)	1/2	1/4	1/8	1/16
‖u-u_CN‖_b	3.068 6E-3	6.695 7E-4	1.617 0E-4	4.003 1E-5
rate log₂(k)		2.196 3	2.049 9	2.014 2
‖u-u_ST‖_b	7.529 1E-6	7.082 6E-7	8.170 7E-8	9.852 2E-9
rate log₂(k)		3.410 1	3.115 7	3.052 0
‖p-p_CN‖_b	3.538 7E-3	5.569 8E-4	1.221 2E-4	1.880 6E-5
rate log₂(k)		2.667 5	2.189 3	2.699 0
‖p-p_ST‖_b	7.511 3E-5	8.248 3E-6	7.801 6E-7	7.875 8E-8
rate log₂(k)		3.186 9	3.402 3	3.308 2

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