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On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation

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Summary.

It is shown in this paper that the optimal approximation property of bilinear or trilinear finite elements would be retained when the affine mapping is used to replace the Q1 mapping on each element, if the grids are refined nestedly. The new method truncates the quadratic and cubic terms in reference mappings and produces constant Jacobians and Jacobian matrices. This would avoid a shortcoming of the quadrilateral and hexahedral elements where the integrals of rational functions have to be computed or approximated. Numerical tests verify the analysis.

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Acknowledgments.

The author thanks a referee who carefully read this manuscript and made it much more rigorous in notations, statements and proofs.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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  1. Shangyou Zhang
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Correspondence to Shangyou Zhang.

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Mathematics Subject Classification (2000): 65N30, 65N50, 65N55

Revised version received January 29, 2004

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Zhang, S. On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation. Numer. Math. 98, 559–579 (2004). https://doi.org/10.1007/s00211-004-0536-7

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  • DOI: https://doi.org/10.1007/s00211-004-0536-7

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