抛物积分微分方程的Wilson元收敛性分析

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.

图/表 4

表 1

$t=0.5$ 时的数值计算结果( $\tau=h^2)$ "

$m\times m$	$\left\\| {u-u_h } \right\\|_h$	收敛阶	$\left\\| {u-u_h } \right\\|_0$	收敛阶
$4\times 4$	0.023734626	---	0.000995509	---
$8\times 8$	0.006276340	1.9190	0.000187996	2.4047
$16\times 16$	0.001559389	2.0089	0.000043305	2.1181
$32\times 32$	0.000386405	2.0128	0.000010702	2.0167

表 1

表 2

$t=1$ 时的数值计算结果( $\tau=h^2)$ "

$m\times m$	$\left\\| {u-u_h } \right\\|_h$	收敛阶	$\left\\| {u-u_h } \right\\|_0$	收敛阶
$4\times 4$	0.039374206	---	0.001662729	---
$8\times 8$	0.010324365	1.9312	0.000341116	2.2852
$16\times 16$	0.002554045	2.0152	0.000082887	2.0410
$32\times 32$	0.000631856	2.0151	0.000020794	1.9950

表 2

图 1

误差图( $t=0.5$ )"

图 1

图 2

误差图( $t=1$ )"

图 2

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