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Convergent Adaptive Finite Element Method Based on Centroidal Voronoi Tessellations and Superconvergence

Published online by Cambridge University Press:  20 August 2015

Yunqing Huang ,
Hengfeng Qin ,
Desheng Wang  and
Qiang Du
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China
Hengfeng Qin*
Affiliation:
Department of Mechanical and Electronic Engineering, School of Mechanical Engineering, Xiangtan University, Hunan 411105, China
Desheng Wang*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
Qiang Du*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
*
Corresponding author.Email:huangyq@xtu.edu.cn
Email:qinhf2001@yahoo.com.cn
Email:desheng@ntu.edu.sg
Email:qdu@math.psu.edu
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Abstract

We present a novel adaptive finite element method (AFEM) for elliptic equations which is based upon the Centroidal Voronoi Tessellation (CVT) and superconvergent gradient recovery. The constructions of CVT and its dual Centroidal Voronoi Delaunay Triangulation (CVDT) are facilitated by a localized Lloyd iteration to produce almost equilateral two dimensional meshes. Working with finite element solutions on such high quality triangulations, superconvergent recovery methods become particularly effective so that asymptotically exact a posteriori error estimations can be obtained. Through a seamless integration of these techniques, a convergent adaptive procedure is developed. As demonstrated by the numerical examples, the new AFEM is capable of solving a variety of model problems and has great potential in practical applications.

Keywords

65N5065N15Finite element methodssuperconvergent gradient recoveryCentroidal Voronoi Tessellationadaptive methods
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 2 , August 2011 , pp. 339 - 370
Copyright
Copyright © Global Science Press Limited 2011

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References

[1] Ainsworth, M. and Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis, Wiley Interscience, New York, 2000.Google Scholar
[2] Babuška, I. and Strouboulis, T., The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation, Oxford Science Publications, 2001.CrossRefGoogle Scholar
[3] Babuška, I. and Miller, A., A feedback finite element method with a posteriori error estimation: part I, the finite element method and some basis properties of the a posteriori error estimator, Comput. Meth. Appl. Mech. Engrg., 61 (1987), 1–40.CrossRefGoogle Scholar
[4] Babuška, I. and Rhcinboldt, W. C., A posteriori error estimates for the finite element method, Int. J. Numer. Method. Engrg., 12 (1978), 1597–1615.CrossRefGoogle Scholar
[5] Bank, R. E. and Smith, R. K., Mesh smoothing using a posterior error estimates, SIAM J. Numer. Anal., 34 (1997), 979–997.CrossRefGoogle Scholar
[6] Bank, R. E. and Xu, J. C., Asymptotically exact a posterior error estimates, part I: grids with superconvergence, SIAM J. Numer. Anal., 41 (2003), 2294–2312.Google Scholar
[7] Bank, R. E., PLTMG: A software package for solving elliptic partial differential equations users guide 10.0, department of mathematics, University of California at San Diego, 2007.Google Scholar
[8] Bernardi, C. and Verfürth, R., Adaptive finite element methods for elliptic equations with nonsmooth coefficients, Numer. Math., 85 (2000), 579–608.Google Scholar
[9] Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1996.Google Scholar
[10] Chen, Z. M. and Dai, S. B., On the efficiency of adaptive finite element methods for elliptic problems with discountinous coefficients, SIAM J. Sci. Comput., 24 (2002), 443–462.CrossRefGoogle Scholar
[11] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[12] Chen, C. M. and Huang, Y. Q., High Accuracy Theory of Finite Element Methods, Hunan Science Press, Hunan, China, 1995 (in Chinese).Google Scholar
[13] Du, Q., Faber, V. and Gunzburger, M., Centroidal Voronoi Tessellations: applications and algorithms, SIAM Rev., 41 (1999), 637–676.Google Scholar
[14] Du, Q. and Gunzburger, M., Grid generation and optimization based on Centroidal Voronoi Tessellations, Appl. Comput. Math., 133 (2002), 591–607.Google Scholar
[15] Du, Q. and Wang, D. S., Tetrahedral mesh generation and optimization based on Centroidal Voronoi Tessellations, Int. J. Numer. Method. Engrg., 56 (2003), 1355–1373.CrossRefGoogle Scholar
[16] Du, Q. and Wang, D., On the optimal Centroidal Voronoi Tessellations and the Gersho’s conjecture in the three dimensional space, Int. J. Comput. Math. Appl., 499 (2005), 1355–1373.Google Scholar
[17] Du, Q. and Wang, D., Anisotropic Centroidal Voronoi Tessellations and their applications, SIAM. J. Sci. Comput., 26 (2005), 737–761.Google Scholar
[18] Du, Q. and Wang, D., Mesh optimization based on Centroidal Voronoi Tessellation, Int. J. Numer. Anal. Model., 2 (2005), 100–114.Google Scholar
[19] Du, Q. and Ju, L., Numerical simulation of the quantized vortices on a thin superconducting hollow sphere, J. Comput. Phys., 201 (2004), 511–530.Google Scholar
[20] Douglas, J. J. and Dupont, T., Superconvergence for Galerkin methods for the two-point boundary problem via local projections, Numer. Math., 21 (1973), 270–278.Google Scholar
[21] Du, Q. and Ju, L., Finite volume methods on spheres and spherical centroidal Voronoi meshes, SIAM J. Numer. Anal., 43 (2005), 1673–1692.CrossRefGoogle Scholar
[22] Du, Q. and Wang, D., Constrained boundary recovery for 3D Delaunay triangulations, Int. J. Numer. Method. Engrg., 61 (2004), 1471–1500.CrossRefGoogle Scholar
[23] George, P. L. and Borouchaki, H., Delaunay Triangulation and Meshing, Application to Finite Elements Methods, Hermès, Paris, 1998.Google Scholar
[24] Zhou, A. and Lin, Q., Optimal and superconvegence estimates of the finite element method for a scalar hyperbolic equation, Acta. Math. Sci., (English Ed.) 14 (1994), 90–94.Google Scholar
[25] Yan, N. and Zhou, A., Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes, Comput. Methods. Appl. Mech. Engrg., 190 (2001), 4289–4299.Google Scholar
[26] Goodsell, G. and Whiteman, J. R., A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations, Int. J. Num. Method. Engrg., 27 (1989), 469–481.Google Scholar
[27] Huang, Y., Qin, H. and Wang, D., Centroidal Voronoi tessellation-based finite element superconvergence, Int. J. Numer. Method. Engrg., 76 (2008), 1819–1839.Google Scholar
[28] Ju, L., Gunzburger, M. and Zhao, W. D., Adaptive finite element methods for elliptic PDEs based on conforming centroidal Voronoi Delaunay triangulations, SIAM J. Sci. Comput., 28 (2006), 2023–2053.Google Scholar
[29] Křížek, M., Superconvergence phenomena in the finite element method, Comput. Methods. Appl. Mech. Engrg., 116 (1994), 157–163.Google Scholar
[30] Kellogg, R. B., On the Poisson equation with intersecting interface, Appl. Anal., 4 (1975), 101–129.Google Scholar
[31] Levine, N. D., Superconvergent recovery of the gradient from piecewise linear finite element approximations, IMA J. Numer. Anal., 5 (1985), 407–427.Google Scholar
[32] Lakhany, A. M., Marek, I. and Whiteman, J. R., Superconvergence results on midly structured triangulations, Comput. Methods. Appl. Mech. Engrg., 189 (2000), 1–75.Google Scholar
[33] Lin, R. and Zhang, Z., Natural superconvergent points of triangular finite elements, Numer. Meth. PDEs., 20 (2004), 864–906.Google Scholar
[34] Lloyd, S., Least square quantization in PCM, IEEE T. Infom. Theory., 28 (1982), 129–137.Google Scholar
[35] Morin, P., Nochetto, R. H. and Siebert, K. G., Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 37 (2000), 466–488.Google Scholar
[36] Naga, A. and Zhang, Z. M., A posteriori error estimates based on the polynomial preserving recovery, SIAM J. Numer. Anal., 42 (2004), 1780–1800.CrossRefGoogle Scholar
[37] Naga, A. and Zhang, Z. M., The polynomial-preserving recovery for higher order finite element methods in 2D and 3D, Dis. Cont. Dyn. B., 53 (2005), 769–798.Google Scholar
[38] Verfürth, R., A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, New York, Wiley-Teubner, 1996.Google Scholar
[39] Wahlbin, L. B., Superconvergence in Galerkin Finite Element Methods, Volume 1605 of Lecture Notes in Mathematics, Springer, Berlin, 1995.Google Scholar
[40] Wang, D., Hassan, O., Weatherill, N. and Morgan, K., Enhanced remeshing from STL files and its application to surface grid generation, Commun. Numer. Methods. Engrg., 23 (2007), 227–239.Google Scholar
[41] Watson, D., Computing the n-dimensional Delaunay tessellation with applications to Voronoi polytopes, Comput. J., 24 (1981), 167–172.Google Scholar
[42] Wu, H. and Zhang, Z. M., Can we have superconvergent gradient recovery under adaptive meshes, SIAM J. Numer. Anal., 45 (2007), 1701–1722.Google Scholar
[43] Xu, J. C. and Zhang, Z. M., Analysis of recovery types a posteriori estimators for middly structured grids, Math. Comput., 73 (2004), 1139–1152.Google Scholar
[44] Zhu, J. Z. and Zienkiewwicz, O. C., Superconvergence recovery technique and a posteriori error estimators, Int. J. Numer. Method. Engrg., 30 (1990), 1321–1339.Google Scholar
[45] Zienkiewicz, O. C. and Zhu, J. Z., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Method. Engrg., 24 (1987), 337–357.Google Scholar
[46] Zienkiewicz, O. C. and Zhu, J. Z., The superconvergent patch recovery (SPR) and adaptive finite element refinement, Comput. Methods. Appl. Mech. Engrg., 101 (1992), 207–224.Google Scholar