##
**Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order.**
*(English)*
Zbl 1231.65122

Summary: Based on the newly-developed element energy projection (EEP) method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method (FEM) is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and 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 |

### Keywords:

finite element method (FEM); self-adaptive solution; super-convergence; optimal convergence order; element energy projection; condensed shape functions### Software:

COLSYS
PDFBibTeX
XMLCite

\textit{S. Yuan} et al., Appl. Math. Mech., Engl. Ed. 29, No. 5, 591--602 (2008; Zbl 1231.65122)

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 super-convergence 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: Shanghai Science and Technology Press, 1994 (in Chinese). |

[5] | Chen Chuanmiao. Constructive theory of super-convergence in FEM[M]. Changsha: Hunan Science and Technology Press, 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]. Beijing-New York: Science Press, 1993. · Zbl 0891.65123 |

[8] | Yuan Si. The loss and recovery of stress accuracy in FEM as seen from matrix displacement method[J]. Mechanics in Engineering, 1998, 20(4):1–6 (in Chinese). |

[9] | Yuan Si, Wang Mei. An element-energy-projection method for post-computation of superconvergent solutions in one-dimensional FEM[J]. Engineering Mechanics, 2004, 21(2):1–9 (in Chinese). |

[10] | Yuan Si, Wang Mei, He Xuefeng. Computation of super-convergent solutions in one-dimensional C 1 FEM by EEP method[J]. Engineering Mechanics, 2006, 23(2):1–9 (in Chinese). |

[11] | Wang Mei, Yuan Si. Computation of super-convergent nodal stresses of Timoshenko beam elements by EEP method[J]. Applied Mathematics and Mechanics (English Edition), 2004, 25(11):1228–1240. DOI 10.1007/BF02438278 · Zbl 1147.74401 |

[12] | Yuan Si, Lin Yongjing. An EEP method for post-computation of super-convergent solutions in one-dimensional Galerkin FEM for second order non-self-adjoint boundary-value problem[J]. Chinese Journal of Computational Mechanics, 2007, 24(2):142–147 (in Chinese). |

[13] | Yuan Si, Wang Mei, Wang Xu. An EEP method for super-convergent solutions in 2D FEMOL[J]. Engineering Mechanics, 2007, 24(1):1–10 (in Chinese). |

[14] | Yuan Si, He Xuefeng. A self-adaptive strategy for one-dimensional FEM based on EEP method[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(11):1461–1474. DOI 10.1007/s10483-006-1103-1 · Zbl 1231.65121 |

[15] | Zhao Qinghua, Zhou Shuzi, Zhu Qiding. Mathematical analysis of EEP method for one-dimensional finite element postprocessing[J]. Applied Mathematics and Mechanics (English Edition), 2007, 28(4):441–445. DOI 10.1007/s 10483-007-0403-y · Zbl 1231.65225 |

[16] | Yuan Si, Wang Xu, Xing Qinyan, Ye Kangsheng. A scheme with optimal order of super-convergence based on the element energy projection method: I formulation[J]. Engineering Mechanics, 2007, 24(10):1–5 (in Chinese). |

[17] | Yuan Si, Xing Qinyan, Wang Xu, Ye Kangsheng. A scheme with optimal order of super-convergence based on the element energy projection method: II numerical results[J]. Engineering Mechanics, 2007, 24(11):1–5 (in Chinese). |

[18] | Yuan Si, Zhao Qinghua. A scheme with optimal order of super-convergence based on the element energy projection method: III mathematical analysis[J]. Engineering Mechanics, 2007, 24(12):1–6 (in Chinese). |

[19] | Douglas J, Dupont T. Galerkin approximations for the two point boundary problems using continuous piecewise polynomial spaces[J]. Numerical Mathematics, 1974, 22(2):99–109. · Zbl 0331.65051 |

[20] | Strang G, Fix G. An analysis of the finite element method[M]. London: Prentice-Hall, 1973. · Zbl 0356.65096 |

[21] | Wang Xu. Adaptive analysis of 1D FEM and 2D FEMOL based on EEP super-convergent method[D]. Ph D Dissertation. Beijing: Tsinghua University, 2007 (in Chinese). |

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.