Journal of Scientific Computing

, Volume 59, Issue 1, pp 53–79

Constraint-Free Adaptive FEMs on Quadrilateral Nonconforming Meshes

Authors

    • LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Academy of Mathematics and Systems ScienceChinese Academy of Sciences
    • Beijing Computational Science Research Center
  • Zhong-Ci Shi
    • LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Academy of Mathematics and Systems ScienceChinese Academy of Sciences
  • Qiang Du
    • Department of MathematicsPennsylvania State University
    • Beijing Computational Science Research Center
Article

DOI: 10.1007/s10915-013-9753-5

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

Abstract

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.

Keywords

Adaptive finite elementA posteriori error estimate QuadrilateralNonconformingHanging nodeConstraint-free

Mathematics Subject Classification

65N1265N1565N3065N5035J25

Copyright information

© Springer Science+Business Media New York 2013