New error estimates of biquadratic Lagrange elements for Poisson's equation

doi:10.1016/j.apnum.2005.06.003

	New error estimates of biquadratic Lagrange elements for Poisson's equation
	Huang, HT ; Li, ZC ; Zhou, AH
	2006-05-01
发表期刊	APPLIED NUMERICAL MATHEMATICS
ISSN	0168-9274
卷号	56 期号:5 页码:712-744
摘要	In this paper, we report some new ultraconvergence results of biquadratic Lagrange elements for the Dirichlet problem of Poisson's equation, -Delta u = f. The point-line-area interpolant in [V Girault, P.A. Raviart, A Finite Element Methods for Navier-Stokes Equation, Theory and Algorithms, Springer, 1986] is chosen in this paper, instead of the traditional pure point interpolant in [P.G. Ciarlet, Basic error estimates for elliptic problems, in: P.G. Ciarlet, J.L. Lions (Eds.), Finite Element Methods, Part 1, North-Holland, Amsterdam, 199 1, pp. 17-351]. Suppose that the solution is smooth enough, by means of an a posteriori interpolant, the ultraconvergence O(h(4)) in H I norm is proved for uniform rectangles rectangle(ij), and the higher ultraconvergence O(h(6-l)) in H-l (l = 0, 1) norm under the special case of uniform squares rectangle(ij) and f(xxyy) = 0. Even when f(xxyy) not equal 0, we propose two techniques: (1) the Richardson extrapolation method and (2) the correction method, to retain the same higher ultraconvergence results. Moreover, the ultraconvergence O(h(6-l) vertical bar 1nh vertical bar) is also proved for l (l = 0, 1) order infinite norms. In this paper, the numerical experiments are provided to validate all the ultraconvergence results made. Note that the new ultraconvergence results under the special case are three order higher than the optimal convergence rate in [P.G. Ciarlet, Basic error estimates for elliptic problems, in: P.G. Ciarlet, J.L. Lions (Eds.), Finite Element Methods, Part 1, North-Holland, Amsterdam, 1991, pp. 17=351], and one order than that in [Q. Lin, N. Yan, A. Zhou, A rectangle test for interpolated finite elements, in: Proc. Sys. Sci. and Sys. Engrg., Great Wall Culture Publishers, Hong Kong, 1991, pp. 217-229].
关键词	lagrange elements biquadratic elements Poisson's equation global superconvergence ultraconvergence
DOI	10.1016/j.apnum.2005.06.003
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000236426000008
出版者	ELSEVIER SCIENCE BV
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/2645
专题	计算数学与科学工程计算研究所
通讯作者	Li, ZC
作者单位	1.Natl Sun Yat Sen Univ, Dept Appl Math, Kaohsiung 80424, Taiwan 2.Natl Sun Yat Sen Univ, Dept Comp Sci & Engn, Kaohsiung 80424, Taiwan 3.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, Beijing 100080, Peoples R China
推荐引用方式 GB/T 7714	Huang, HT,Li, ZC,Zhou, AH. New error estimates of biquadratic Lagrange elements for Poisson's equation[J]. APPLIED NUMERICAL MATHEMATICS,2006,56(5):712-744.
APA	Huang, HT,Li, ZC,&Zhou, AH.(2006).New error estimates of biquadratic Lagrange elements for Poisson's equation.APPLIED NUMERICAL MATHEMATICS,56(5),712-744.
MLA	Huang, HT,et al."New error estimates of biquadratic Lagrange elements for Poisson's equation".APPLIED NUMERICAL MATHEMATICS 56.5(2006):712-744.