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Mesh Quality and More Detailed Error Estimates of Finite Element Method

  • Yunqing Huang (a1), Liupeng Wang (a1) and Nianyu Yi (a1)

Abstract

In this paper, we study the role of mesh quality on the accuracy of linear finite element approximation. We derive a more detailed error estimate, which shows explicitly how the shape and size of elements, and symmetry structure of mesh effect on the error of numerical approximation. Two computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the mesh. In compare with the standard a priori error estimates, which only yield information on the asymptotic error behaviour in a global sense, our proposed error estimate considers the effect of local element geometry properties, and is thus more accurate. Under certain conditions, the traditional error estimates and supercovergence results can be derived from the proposed error estimate. Moreover, the estimators Ge and Gv are computable and thus can be used for predicting the variation of errors. Numerical tests are presented to illustrate the performance of the proposed parameters Ge and Gv .

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Corresponding author

*Corresponding author. Email addresses: huangyq@xtu.edu.cn (Y. Q. Huang), wangliupeng@xtu.edu.cn (L. P. Wang), yinianyu@xtu.edu.cn (N. Y. Yi)

References

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[1] Babuska, I. and Aziz, A., On the angle condition in the finite element method, SIAM J. Numer. Anal., 13 (1976), pp. 214226.
[2] Chen, C., Optimal points of the stresses for triangular linear element, Numer. Math. J. Chinese Univ., 2 (1980), pp. 1220(in Chinese).
[3] Chen, C., Superconvergence of finite element solutions and their derivatives, Numer. Math. J. Chinese Univ., 3 (1981), pp. 118125(in Chinese).
[4] Chen, C., Superconvergence of finite element approximations to nonlinear elliptic problems, (Proc. China-France Sympos. on Finite Element Methods, Beijing, 1982), Gordon and Breach, New York, pp. 622640, 1983.
[5] Chen, C. and Huang, Y., High Accuracy Theory of Finite Element Methods, Hunan Science Press, Hunan, China(in Chinese), 1995.
[6] Du, Q., Faber, V. and Gunzburger, M., Centroidal Voronoi tessellations: applications and algorithms, SIAM Rev., 41 (1999), pp. 637676.
[7] Huang, Y., Qin, H. and Wang, D., Centroidal Voronoi tessellation-based finite element superconvergence, Int. J. Numer. Methods Eng., 76 (2008), pp. 18191839.
[8] Huang, Y., Li, J., Wu, C. and Yang, W., Superconvergence analysis for linear tetrahedral edge elements, J. Sci. Comput., 62 (2015), pp. 122145.
[9] Huang, Y. and Xu, J., Superconvergence of quadratic finite elements on mildly structured grids, Math. Comput., 77 (2008), pp. 12531268.
[10] Křižek, M., Superconvergence results for linear triangular elements, Lecture Notes in Mathematics, 1192, pp. 315320, Springer, 1986.
[11] Křižek, M., Neittaanmäki, P. and Stenberg, R., Finite element methods: superconvergence, post-processing, and a posteriori estimates, Lecture Notes in Pure and Applied Mathematics Series, Vol. 196, Marcel Dekker, New York, 1997.
[12] Lakhany, A. M., Marek, I. and Whiteman, J. R., Superconvergence results in mildly structured triangulations, Comput. Methods Appl. Mech. Engry, 189 (2000), pp. 175.
[13] Lin, Q. and Yan, N., The Construction and Analysis of High Efficiency Finite Elements, Hebei University Press, Hebei, China(in Chinese), 1996.
[14] Persson, P. and Strang, G., A simple mesh generator in Matlab, SIAM Rev., 46 (2004), pp. 329345.
[15] Wahlbin, L. B., Superconvergence in Galkerkin Finite Element Methods, Springer Verlag, Berlin, 1995.
[16] Yi, N. Y., A posteriori Error Estimates Based on Gradient Recovery and Adaptive Finite Element Methods, Ph.D. thesis, Xiangtan University, 2011.
[17] Zhu, Q., Natural inner superconvergence for the finite element method, (Proc. China-France Sympos. on Finite Element Methods, Beijing, 1982), Gordon and Breach, New York, pp. 935960, 1983.
[18] Zhu, Q. and Lin, Q., Finite Element Superconvergence Theory, Hunan Science Press, Hunan, China(in Chinese), 1989.
[19] Zlámal, M., On the finite element method, Numer. Math., 12 (1968), pp. 394409.
[20] Zlámal, M., Superconvergence and reduced integration in the finite element method, Math. Comput., 32 (1978), pp. 663685.

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Mesh Quality and More Detailed Error Estimates of Finite Element Method

  • Yunqing Huang (a1), Liupeng Wang (a1) and Nianyu Yi (a1)

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