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.

参考文献

1	Hayes L J . A modified backward time discretization for nonlinear parabolic equations using patch approximations. SIAM J Numer Anal, 1981, 18: 781- 793
2	Rachford H , J r . Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations. SIAM J Numer Anal, 1973, 10 (6): 1010- 1026
3	Luskin M . A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions. SIAM J Numer Anal, 1979, 16 (2): 284- 299
4	Thomée V . Galerkin Finite Element Methods for Parabolic Problems. Berlin: Springer, 2000
5	陈传军, 张晓艳, 赵鑫. 一维非线性抛物问题两层网格有限体积元逼近. 数学物理学报, 2017, 37A (5): 962- 975
5	Chen C , Zhang X , Zhao X . A two-grid finite volume element approximation for one-dimensional nonlinear parabolic equations. Acta Math Sci, 2017, 37A (5): 962- 975
6	Shi D , Wang J , Yan F . Unconditional superconvergence analysis for nonlinear parabolic equation with EQ₁^rot nonconforming finite element. J Sci Comput, 2017, 70 (1): 85- 111
7	Li D , Wang J . Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system. J Sci Comput, 2017, 72 (2): 892- 915
8	Pani A K . An H¹-Galerkin mixed finite element methods for parabolic partial differential equatios. SIAM J Numer Anal, 1998, 35 (2): 712- 727
9	石东洋, 史艳华. 半线性伪双曲方程最低阶的H¹-Galerkin混合元方法. 系统科学与数学, 2015, 35 (5): 514- 526
9	Shi D , Shi Y . The lowest order H¹-Galerkin mixed finite element method for semi-linear pseudo-hyperbolic equation. J Syst Sci Complex, 2015, 35 (5): 514- 526
10	Liu Y , Li H . H¹-Galerkin mixed finite element methods for pseudo-hyperbolic equations. Appl Math Comput, 2009, 212 (2): 446- 457
11	Shi D , Wang J . Superconvergence analysis of an H¹-Galerkin mixed finite element method for Sobolev equations. Comput Math Appl, 2016, 72 (6): 1590- 1602
12	Shi D , Wang J . Unconditional superconvergence analysis of an H¹-galerkin mixed finite element method for nonlinear Sobolev equations. Numer Methods Partial Differential Equations, 2018, 34 (1): 145- 166
13	Shi D , Liao X , Tang Q . Highly efficient H¹-Galerkin mixed finite element method (MFEM) for parabolic integro-differential equatios. Appl Math Mech Engl Ed, 2014, 35 (7): 897- 912
14	Lin Q , Lin J F . Finite Element Methods:Accuracy and Improvement. Beijing: Science Press, 2006
15	陈传淼, 黄云清. 有限元高精度理论. 长沙: 湖南科学技术出版社, 1995
15	Chen C , Huang Y . High Accuracy Theory of Finite Element Method. Changsha: Hunan Science and Technology Press, 1995
16	陈艳萍, 鲁祖亮. 最优控制问题高效高精度算法. 北京: 科学出版社, 2015
16	Chen Y , Lu Z . High Efficient and Accuracy Numerical Methods for Optimal Control Probelems. Beijing: Science Press, 2015

[1]	陈友朋, 谢春红. 一类带非线性边界条件的非线性抛物方程的解的整体存在性[J]. 数学物理学报, 2005, 25(6): 881-889.
[2]	穆春来. 关于具有非线性边界条件的非线性抛物型方程Blow-up现象[J]. 数学物理学报, 1996, 16(S1): 126-132.

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

RichHTML

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

Metrics

本文评价

推荐阅读 0