Commun. Comput. Phys., 10 (2011), pp. 339-370.


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

Yunqing Huang 1*, Hengfeng Qin 2, Desheng Wang 3, Qiang Du 4

1 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China.
2 Department of Mechanical and Electronic Engineering, School of Mechanical Engineering, Xiangtan University, Hunan 411105, China.
3 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore.
4 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.

Received 3 February 2010; Accepted (in revised version) 5 November 2010
Available online 27 April 2011
doi:10.4208/cicp.030210.051110a

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.

AMS subject classifications: 65N50, 65N15

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Key words: Finite element methods, superconvergent gradient recovery, Centroidal Voronoi Tessellation, adaptive methods.

*Corresponding author.
Email: huangyq@xtu.edu.cn (Y. Huang), qinhf2001@yahoo.com.cn (H. Qin), desheng@ntu.edu.sg (D. Wang), qdu@math.psu.edu (Q. Du)
 

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