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H 1-interpolation on quadrilateral and hexahedral meshes with hanging nodes

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Abstract

We propose a Scott-Zhang type finite element interpolation operator of first order for the approximation of H 1-functions by means of continuous piecewise mapped bilinear or trilinear polynomials. The novelty of the proposed interpolation operator is that it is defined for general non-affine equivalent quadrilateral and hexahedral elements and so-called 1-irregular meshes with hanging nodes. We prove optimal local approximation properties of this interpolation operator for functions in H 1. As necessary ingredients we provide a definition of a hanging node and a rigorous analysis of the issue of constrained approximation which cover both the two- and three-dimensional case in a unified fashion.

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References

  • Ainsworth M. and Oden J. T. (2000). A posteriori error estimation in finite element analysis. Pure and Applied Mathematics. Wiley, Chichester

    Google Scholar 

  • Bernardi C. and Girault V. (1998). A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35(5): 1893–1916

    Article  MATH  Google Scholar 

  • Brenner, S. C., Scott, R.: The mathematical theory of finite element methods. Springer 1994.

  • Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, vol. 15. Springer 1991.

  • Chou S.-H. and He S. (2002). On the regularity and uniformness conditions on quadrilateral grids. Comput. Meth. Appl. Mech. Engng. 191(45): 149–5158

    Article  Google Scholar 

  • Ciarlet P. G. (1978). The finite element method for elliptic problems. Studies in Mathematics and its Applications, vol. 4. North-Holland, Amsterdam New York Oxford

    Google Scholar 

  • Clément Ph. (1975). Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9(R-2): 77–84

    Google Scholar 

  • Demkowicz L., Oden J. T., Rachowicz W. and Hardy O. (1989). Toward a universal h-p adaptive finite element strategy. I: Constrained approximation and data structure. Comput. Meth. Appl. Mech. Engng. 77(1/2): 79–112

    Article  Google Scholar 

  • Dupont T. and Scott R. (1980). Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34: 441–463

    Article  MATH  Google Scholar 

  • Girault V. and Raviart P.-A. (1986). Finite element methods for Navier–Stokes equations. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Heuveline V. and Rannacher R. (2001). A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15(1–4): 107–138

    Article  MATH  Google Scholar 

  • Heuveline V. and Rannacher R. (2003). Duality-based adaptivity in the hp-finite element method. J. Numer. Math. 11(2): 95–113

    Article  MATH  Google Scholar 

  • Heuveline, V., Schieweck, F.: Constrained H 1-interpolation on quadrilateral and hexadral meshes with hanging nodes. IWRMM-Preprint 05/02, Universität Karlsruhe, Fakultät für Mathematik, 2005. http://www-ian.math.uni-magdeburg.de/schiewec/papers.html.

  • Heuveline V. and Schieweck F. (2007). On the inf-sup condition for higher order mixed fem on meshes with hanging nodes. Math. Modell. Numer. Anal. 41(1): 1–20

    Article  Google Scholar 

  • Matthies G. and Tobiska L. (2002). The inf-sup condition for the mapped Q k -P k-1 disc element in arbitrary space dimensions. Computing 69: 119–139

    Article  MATH  Google Scholar 

  • Ming P. and Shi Z. C. (2002). Quadrilateral mesh revisited. Comput. Meth. Appl. Mech. Engng. 191(49–50): 5671–5682

    Article  MATH  Google Scholar 

  • Schieweck, F.: A general transfer operator for arbitrary finite element spaces. Preprint 25/00, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik, 2000. http://www-ian.math.uni-magdeburg.de/~schiewec.

  • Schötzau S., Schwab C. and Stenberg R. (1999). Mixed hp-fem on anisotropic meshes ii: Hanging nodes and tensor products of boundary layer meshes. Numer. Math. 83: 667–697

    MATH  Google Scholar 

  • Scott L. R. and Zhang S. (1990). Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190): 483–493

    Article  MATH  Google Scholar 

  • Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner Series in Advances in Numerical Mathematics. Wiley-Teubner 1996.

  • Verfürth R. (1999). Error estimates for some quasi-interpolation operators. Math. Model. Numer. Anal. 33(4): 695–713

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Rechenzentrum und Institut für Angewandte Mathematik, Universität Karlsruhe, Zirkel 2, 76128, Karlsruhe, Germany

    V. Heuveline

  2. Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, 39016, Magdeburg, Germany

    F. Schieweck

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  1. V. Heuveline
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  2. F. Schieweck
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Correspondence to V. Heuveline.

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Heuveline, V., Schieweck, F. H 1-interpolation on quadrilateral and hexahedral meshes with hanging nodes. Computing 80, 203–220 (2007). https://doi.org/10.1007/s00607-007-0233-3

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  • DOI: https://doi.org/10.1007/s00607-007-0233-3

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