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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.

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Notes

  1. Here we do not consider the adaptive moving-mesh method, which can be used on quadrilateral and hexahedral meshes without any hanging nodes.

  2. For the classical constrained FEMs on nonconforming meshes, there are two approaches to assemble stiffness matrix. One is assembling a saddle matrix through an elementwise computation which is based the original bases functions including bases function on hanging nodes, and then eliminating the constraints in the saddle matrix to reduce to a symmetric matrix (the bilinear form is symmetric). The other one is modifying the bases functions as a“macro-element” on regular nodes (add a half of the basis function on a hanging node to the two bases functions on the corresponding two regular nodes, see (5.6) in [30]) and remove all the bases functions on hanging nodes, and then assembling the stiffness matrix with the modified bases functions on regular nodes, which automatically generates a symmetric matrix (the bilinear form is symmetric). Cite Sect. 5 in [30]. However, the well-used element-wise assembly technique is lost in assembling stiffness matrix in the latter approach if using modified bases functions.

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Acknowledgments

The research was supported by National Nature Science Foundation of China grant 11201462, and National Nature Science Foundation of China grant 11371359, and U.S. Department of Energy grant DE-SC0005346, and U.S. NSF grant DMS-1016073.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Xuying Zhao & Zhong-Ci Shi

  2. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Qiang Du

  3. Beijing Computational Science Research Center, Beijing, 100084, People’s Republic of China

    Xuying Zhao & Qiang Du

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  1. Xuying Zhao
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Correspondence to Xuying Zhao.

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Zhao, X., Shi, ZC. & Du, Q. Constraint-Free Adaptive FEMs on Quadrilateral Nonconforming Meshes. J Sci Comput 59, 53–79 (2014). https://doi.org/10.1007/s10915-013-9753-5

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  • DOI: https://doi.org/10.1007/s10915-013-9753-5

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