一类非线性抛物方程H1-Galerkin混合有限元方法的高精度分析

数学物理学报 ›› 2019, Vol. 39 ›› Issue (4): 894-908.

一类非线性抛物方程H¹-Galerkin混合有限元方法的高精度分析

王俊俊*(),杨晓侠

平顶山学院数学与统计学院河南平顶山 467000

收稿日期:2018-02-28 出版日期:2019-08-26 发布日期:2019-09-11
通讯作者: 王俊俊 E-mail:wjunjun8888@163.com
基金资助:
国家自然科学基金(11671369);平顶山学院博士启动基金(PXY-BSQD-2019001);平顶山学院培育基金(PXY-PYJJ-2019006);the NSFC(11671369);the Doctoral Starting Foundation of PingdingshanUniversity(PXY-BSQD-2019001);the University Cultivation Foundation of Pingdingshan(PXY-PYJJ-2019006)

Superconvergence Analysis of an H¹-Galerkin Mixed Finite Element Method for Nonlinear Parabolic Equation

Junjun Wang*(),Xiaoxia Yang

School of Mathematics and Statistics, Pingdingshan University, Henan Pingdingshan 467000

Received:2018-02-28 Online:2019-08-26 Published:2019-09-11
Contact: Junjun Wang E-mail:wjunjun8888@163.com
Supported by:
国家自然科学基金(11671369);平顶山学院博士启动基金(PXY-BSQD-2019001);平顶山学院培育基金(PXY-PYJJ-2019006);the NSFC(11671369);the Doctoral Starting Foundation of PingdingshanUniversity(PXY-BSQD-2019001);the University Cultivation Foundation of Pingdingshan(PXY-PYJJ-2019006)

摘要/Abstract

摘要：

研究了非线性抛物方程的H¹-Galerkin混合有限元方法.利用双线性元及零阶Raviart-Thomas元，在不提高原始解正则性的前提下，创新性的使用分裂技巧等讨论了半离散格式下和Euler全离散格式下的关于原始变量u的H¹（Ω）模及流量p=▽u的H（div；Ω）模的超逼近性质.数值算例证明了理论的正确性.

关键词: 非线性抛物方程, H¹-Galerkin混合有限元方法, 半离散格式和Euler全离散格式, 超逼近性质

Abstract:

Nonlinear parabolic equation is studied by H¹-Galerkin mixed finite element method. The bilinear element and the zero-order Raviart-Thomas elements are utilized to discuss superclose properties of the original variable u in H¹(Ω) and the flux p=▽u in H(div; Ω) under the semi-discrete scheme and Euler fully-discrete scheme. During the process, the splitting technique is used and the regularity of u and p are not improved. The numerical example confirm the theory.

Key words: Nonlinear parabolic equation, H¹-Galerkin mixed finite element method, The semi-discrete scheme and Euler fully-discrete scheme, Superclose properties

中图分类号:

O242.21

王俊俊,杨晓侠. 一类非线性抛物方程H¹-Galerkin混合有限元方法的高精度分析[J]. 数学物理学报, 2019, 39(4): 894-908.

Junjun Wang,Xiaoxia Yang. Superconvergence Analysis of an H¹-Galerkin Mixed Finite Element Method for Nonlinear Parabolic Equation[J]. Acta mathematica scientia,Series A, 2019, 39(4): 894-908.

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$m\times m$	$G_1$	Order	$G_2$	Order
$10\times10$	1.9233$\times 10^{-2}$	---	3.0654$\times 10^{-4}$	---
$20\times20$	9.5820$\times10^{-3}$	1.0051	8.4333$\times10^{-5}$	1.8619
$40\times40$	4.7867$\times10^{-3}$	1.0013	2.1579$\times10^{-5}$	1.9665

$m\times m$	$G_1$	Order	$G_2$	Order
$10\times10$	2.4810$\times10^{-2}$	---	1.1182$\times10^{-3}$	---
$20\times20$	1.2318$\times10^{-2}$	1.0102	2.8110$\times10^{-4}$	1.9920
$40\times40$	6.1481$\times10^{-3}$	1.0026	7.0349$\times10^{-5}$	1.9985

$m\times m$	$G_1$	Order	$G_2$	Order
$10\times10$	3.1907$\times10^{-2}$	---	1.8301$\times10^{-3}$	---
$20\times20$	1.5823$\times10^{-2}$	1.0119	4.5553$\times10^{-4}$	2.0063
$40\times40$	7.8950$\times10^{-3}$	1.0030	1.1367$\times10^{-4}$	2.0026

$m\times m$	$G_1$	Order	$G_2$	Order
$10\times10$	4.0965$\times10^{-2}$	---	2.4049$\times10^{-3}$	---
$20\times20$	2.0316$\times10^{-2}$	1.0118	5.9393$\times10^{-4}$	2.0176
$40\times40$	1.0137$\times10^{-2}$	1.0029	1.4782$\times10^{-4}$	2.0064

$m\times m$	$H_1$	Order	$H_2$	Order
$10\times10$	6.2935$\times10^{-2}$	---	5.5967$\times10^{-3}$	---
$20\times20$	3.1499$\times10^{-2}$	0.9985	1.5489$\times10^{-3}$	1.8534
$40\times40$	1.5750$\times10^{-2}$	0.9999	3.9834$\times10^{-4}$	1.9591

一类非线性抛物方程H¹-Galerkin混合有限元方法的高精度分析

Superconvergence Analysis of an H¹-Galerkin Mixed Finite Element Method for Nonlinear Parabolic Equation

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$m\times m$	$H_1$	Order	$H_2$	Order
$10\times10$	8.1488$\times10^{-2}$	---	1.2718$\times10^{-2}$	---
$20\times20$	4.0543$\times10^{-2}$	1.0071	3.4356$\times10^{-3}$	1.8882
$40\times40$	2.0236$\times10^{-2}$	1.0025	8.7793$\times10^{-4}$	1.9684

$m\times m$	$H_1$	Order	$H_2$	Order
$10\times10$	1.0559$\times10^{-1}$	---	2.1634$\times10^{-2}$	---
$20\times20$	5.2199$\times10^{-2}$	1.0164	5.8430$\times10^{-3}$	1.8885
$40\times40$	2.6003$\times10^{-2}$	1.0054	1.4939$\times10^{-3}$	1.9676

$m\times m$	$H_1$	Order	$H_2$	Order
$10\times10$	1.3681$\times10^{-1}$	---	3.3263$\times10^{-2}$	---
$20\times20$	6.7216$\times10^{-2}$	1.0253	9.0587$\times10^{-3}$	1.8766
$40\times40$	3.3414$\times10^{-2}$	1.0084	2.3235$\times10^{-3}$	1.9630