×

Self-adaptive strategy for one-dimensional finite element method based on element energy projection method. (English) Zbl 1231.65121

Summary: Based on the newly-developed element energy projection (EEP) method for computation of super-convergent results in one-dimensional finite element method (FEM), the task of self-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy was found to be very simple, rapid, cheap and efficient. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm are described. Representative numerical examples are given to show the effectiveness and reliability of the proposed approach.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations

Software:

COLSYS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Babuska I, Rheinboldt W C. A posteriori error analysis of finite element method for one-dimensional problems [J]. SIAM Journal on Numerical Analysis, 1981, 18(3): 565–589. · Zbl 0487.65060
[2] Zienkiewicz O C, Zhu J Z. The superconvergence patch recovery (SPR) and a posteriori error estimates, part 1: the recovery technique [J]. Int J Num Meth Engng, 1992, 33(7): 1331–1364. · Zbl 0769.73084
[3] Zienkiewicz O C, Zhu J Z. The superconvergence patch recovery (SPR) and a posteriori error estimates, part 2: error estimates and adaptivity [J]. Int J Num Meth Engng, 1992, 33(7):1365–1382. · Zbl 0769.73085
[4] Lin Qun, Zhu Qiding. Theory of Pre-and Post-Processing of FEM[M]. Shanghai Science and Technology Press, Shanghai, 1994 (in Chinese).
[5] Chen Chuanmiao. Constructive Theory of Super-convergence in FEM[M]. Hunan Science and Technology Press, Changsha, 2002 (in Chinese). · Zbl 1066.65112
[6] Ascher U, Christiansen J, Russell R D. Algorithm 569, COLSYS: Collocation software for boundary value ODEs[J]. ACM Trans Math Software, 1981, 7(2): 223–229.
[7] Yuan Si. The Finite Element Method of Lines[M]. Science Press, Beijing-New York, 1993. · Zbl 0891.65123
[8] Yuan Si, Wang Mei, Lin Yongjing, Yuan Zheng. New thoughts in super-convergent computation of stresses in FEM(OL)[J]. Engineering Mechanics (Suppl Issue), Changsha, 2002, 112–118 (in Chinese).
[9] Yuan Si, Wang Mei. An element-energy-projection method for post-computation of super-convergent solutions in one-dimensional FEM [J]. Engineering Mechanics, 2004, 21(2): 1–9 (in Chinese). · Zbl 1147.74401
[10] Strang G, Fix G. An Analysis of the Finite Element Method[M]. Prentice-Hall, London, 1973. · Zbl 0356.65096
[11] Douglas J, Dupont T. Galerkin approximations for the two point boundary problems using continuous piecewise polynomial spaces[J]. Numer Math, 1974, 22(2): 99–109. · Zbl 0331.65051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.