抛物积分微分方程的Wilson元收敛性分析

数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1158-1169.

抛物积分微分方程的Wilson元收敛性分析

梁聪刚¹(),杨晓侠¹(),石东洋²()

¹ 平顶山学院数学与统计学院河南平顶山 467000
² 郑州大学数学与统计学院郑州 450001

收稿日期:2018-01-30 出版日期:2019-10-26 发布日期:2019-11-08
作者简介:梁聪刚, E-mail:cgangl@126.com|杨晓侠, E-mail:13733937766@163.com|石东洋, E-mail:shi_dy@zzu.edu.cn
基金资助:
国家自然科学基金(11671369);河南省科技计划项目(162300410082);平顶山学院培育基金(PXY-PYJJ-2019006)

Convergence Analysis of Wilson Element for Parabolic Integro-Differential Equation

Conggang Liang¹(),Xiaoxia Yang¹(),Dongyang Shi²()

¹ School of Mathematics and Statistics, Pingdingshan University, Henan Pingdingshan 467000
² School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001

Received:2018-01-30 Online:2019-10-26 Published:2019-11-08
Supported by:
the NSFC(11671369);the Science and Technology Planning Foundation of Henan Province(162300410082);the University Cultivation Foundation of Pingdingshan(PXY-PYJJ-2019006)

摘要/Abstract

摘要：

该文利用Wilson元对一类抛物积分微分方程提出了新的半离散和全离散逼近格式.基于单元的性质，通过定义新的双线性型，在不需要外推和插值后处理技术的前提下，分别得到了比传统的H¹-范数更大的模意义下相应的O（h²）阶和O（h²+τ）阶的误差分析结果，比通常的关于Wilson元的误差估计高出一阶.这里，h，τ分别表示空间剖分参数和时间步长.最后，给出了一个数值算例，计算结果验证了理论分析的正确性.

关键词: 抛物积分微分方程, Wilson元, 半离散和全离散格式, 收敛性

Abstract:

In this paper, with the help of the wilson element, new semi-discrete and fullydiscrete schemes are proposed for parabolic integro-differential equation. Based on the properties of the element, through defining a new bilinear form, without using the technique of extrapolation and interpolated postprocessing, in the norm which is stronger than the usual H¹-norm, the convergence results with order O(h²)/O(h²+τ) for the primitive solution are obtained for the corresponding schemes, respectively. The above results are just one order higher than the usual error estimates for the wilson element. Here, h and τ are parameters of the subdivision in space and time step, respectively. Finally, numerical results are provided to confirm the theoretical analysis.

Key words: Parabolic integro-differential equation, Wilson element, Semi-discrete and fulldiscrete schemes, Convergence

中图分类号:

O242.21

梁聪刚,杨晓侠,石东洋. 抛物积分微分方程的Wilson元收敛性分析[J]. 数学物理学报, 2019, 39(5): 1158-1169.

Conggang Liang,Xiaoxia Yang,Dongyang Shi. Convergence Analysis of Wilson Element for Parabolic Integro-Differential Equation[J]. Acta mathematica scientia,Series A, 2019, 39(5): 1158-1169.

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