New error estimates of Adini's elements for Poisson's equation
Huang, HT; Li, ZC; Yan, NN
AbstractIn this paper, we report some new discoveries of Adini's elements for Poisson's equation in error estimates, stability analysis and global superconvergence. It is well known that the optimal convergence rate \\u - u(/1)\\(1) = O(h(3)\u\(4)) can be obtained, where u(/1) and u are the Adini's solution and the true solution, respectively. In this paper, for all kinds of boundary conditions of Poisson's equations, the supercloseness parallel tou(I)(A)-u(/1)parallel to = O(h(3.5)parallel touparallel to(5)) can be obtained for uniform rectangles rectangle(ij), where u A is the Adini's interpolation of the true solution u. Moreover, for the Neumann problems of Poisson's equation, new treatments adding the explicit natural constraints (u(n))(ij) = g(ij) on the boundary are proposed to yield the Adini's solution u(ji)* having supercloseness parallel tou(I)(A) - u* parallel to(I) = O(h(4)parallel touparallel to(5)). Hence, the global superconvergence parallel tou - Pi(5)u(h)(*) = O(h(4)parallel touparallel to(5)) can be achieved, where Pi5u(h)(*) is an a posteriori interpolant of polynomials with order five based on the obtained solution u*. New basic estimates of errors are derived for Adini's elements. Numerical experiments in this paper are also provided to verify the supercloseness and superconvergences, O(h(3.5)) and O(h(4)), and the standard condition number O(h(-2)). It is worthy pointing out that for the Neumann problems on rectangular domains, the traditional finite element method is not as good as the newly proposed method interpolating the Neumann condition in this paper. Not only is the new method more accurate, but also economical in computation, as the discrete system has less unknowns. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.
KeywordAdini's elements Neumann problems Poisson's equation global superconvergence stability analysis
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000221648600003
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Document Type期刊论文
Corresponding AuthorLi, ZC
Affiliation1.Natl Sun Yat Sen Univ, Dept Appl Math, Kaohsiung 80424, Taiwan
2.Acad Sinica, Acad Math & Syst Sci, Inst Syst Sci, Beijing 100080, Peoples R China
Recommended Citation
GB/T 7714
Huang, HT,Li, ZC,Yan, NN. New error estimates of Adini's elements for Poisson's equation[J]. APPLIED NUMERICAL MATHEMATICS,2004,50(1):49-74.
APA Huang, HT,Li, ZC,&Yan, NN.(2004).New error estimates of Adini's elements for Poisson's equation.APPLIED NUMERICAL MATHEMATICS,50(1),49-74.
MLA Huang, HT,et al."New error estimates of Adini's elements for Poisson's equation".APPLIED NUMERICAL MATHEMATICS 50.1(2004):49-74.
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