, Volume 59, Issue 1, pp 53-79
Date: 31 Aug 2013

Constraint-Free Adaptive FEMs on Quadrilateral Nonconforming Meshes

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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.