Journal of Scientific Computing

, Volume 59, Issue 1, pp 53–79

Constraint-Free Adaptive FEMs on Quadrilateral Nonconforming Meshes


DOI: 10.1007/s10915-013-9753-5

Cite this article as:
Zhao, X., Shi, ZC. & Du, Q. J Sci Comput (2014) 59: 53. doi:10.1007/s10915-013-9753-5


Finite element methods (FEMs) on nonconforming meshes have been much studied in the literature. In all earlier works on such methods , some constraints must be imposed on the degrees of freedom on the edge/face with hanging nodes in order to maintain continuity, which make the numerical implementation more complicated. In this paper, we present two FEMs on quadrilateral nonconforming meshes which are constraint-free. Furthermore, we establish the corresponding residual-based a posteriori error reliability and efficiency estimation for general quadrilateral meshes. We also present extensive numerical testing results to systematically compare the performance among three adaptive quadrilateral FEMs: the constraint-free adaptive \(\mathbb Q _1\) FEM on quadrilateral nonconforming meshes with hanging nodes developed herein, the adaptive \(\mathbb Q _1\) FEM based on quadrilateral red-green refinement without any hanging node recently proposed in Zhao et al. (SIAM J Sci Comput 3(4):2099–2120, 2010), and the classical adaptive \(\mathbb Q _1\) FEM on quadrilateral nonconforming meshes with constraints on hanging nodes. Some extensions are also included in this paper.


Adaptive finite elementA posteriori error estimate QuadrilateralNonconformingHanging nodeConstraint-free

Mathematics Subject Classification


Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  3. 3.Beijing Computational Science Research CenterBeijingPeople’s Republic of China