﻿ 四阶抛物型积分微分方程的H1-Galerkin混合元方法 H1-Galerkin Mixed Element Method for Fourth-Order Parabolic Integro-Differential Equation

Vol.05 No.03(2016), Article ID:18296,11 pages
10.12677/AAM.2016.53043

H1-Galerkin Mixed Element Method for Fourth-Order Parabolic Integro-Differential Equation

Yan Li, Yaxin Hou

School of Mathematical Sciences, Inner Mongolia University, Hohhot Inner Mongolia

Received: Jul. 20th, 2016; accepted: Aug. 14th, 2016; published: Aug. 17th, 2016

ABSTRACT

In this paper, an H1-Galerkinmixed element method is considered for one-dimensional fourth-or- der integro-differential equation of parabolic type. According to the characteristics of the considered equation, the three auxiliary variables are introduced, then the original fourth-order problem can be split into the coupled system with first order derivative. Some optimal error estimates for both semi-and fully discrete scheme are proved and the stability for fully discrete system is also derived.

Keywords:Fourth-Order Parabolic Integral Differential Equations, H1-Galerkin Mixed Finite Element Method, Error Estimation, Stability Analysis

1. 引言

(1.1)

H1-Galerkin混合方法首先由Pani在文献 [12] 中针对抛物方程问题提出的一种有效的混合元数值方法，同时他指出该方法具有不必满足著名的LBB相容性条件，混合元空间中的多项式次数可以灵活选取，不受混合空间之间的相互限制，同时得到中间变量和原未知量函数的最优收敛结果。正因于此，国际学者开始对此方法进行不断的研究和发展，并数值求解了很多二阶发展方程问题 [12] - [24] 。直到文献 [23] 的出现，方可将该方法应用于四阶偏微分方程数值求解。

2. 一维问题的H1-Galerkin混合有限元格式

(2.1)

(2.2)

(2.3)

，使得：

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

3. 半离散情形下的误差分析

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

4. 全离散稳定性及误差估计

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

(4.25)

5. 总结

H1-Galerkin Mixed Element Method for Fourth-Order Parabolic Integro-Differential Equation[J]. 应用数学进展, 2016, 05(03): 349-359. http://dx.doi.org/10.12677/AAM.2016.53043

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