Journal of Scientific Computing

, Volume 70, Issue 1, pp 60–84 | Cite as

Optimal Quadrilateral Finite Elements on Polygonal Domains

  • Hengguang Li
  • Qinghui ZhangEmail author


We propose three quadrilateral mesh refinement algorithms to improve the convergence of the finite element method approximating the singular solutions of elliptic equations, which are due to the non-smoothness of the domain. These algorithms result in graded meshes consisting of convex and shape-regular quadrilaterals. With analysis in weighted spaces, we provide the selection criteria for the grading parameter, such that the optimal convergence rate can be recovered for the associated finite element approximation. Various numerical tests verify the theory. In addition to the bi-k elements, we also investigate the serendipity elements on the graded quadrilateral meshes in the numerical experiments.


Corner singularity Finite element Graded quadrilateral mesh Error analysis 

Mathematics Subject Classification

65N30 65N50 65N15 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.Guangdong Provincial Key Laboratory of Computational Science and School of Data and Computer ScienceSun Yat-Sen UniversityGuangzhouPeople’s Republic of China

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