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Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness property

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Abstract

We consider a natural class of composite finite elements that provide the mth-order smoothness of the resulting piecewise polynomial function on a triangulated domain and do not require any information on neighboring elements. It is known that, to provide a required convergence rate in the finite element method, the “smallest angle condition” must be often imposed on the triangulation of the initial domain; i.e., the smallest possible values of the smallest angles of the triangles must be lower bounded. On the other hand, the negative role of the smallest angle can be weakened (but not eliminated completely) by choosing appropriate interpolation conditions. As shown earlier, for a large number of methods of choosing interpolation conditions in the construction of simple (noncomposite) finite elements, including traditional conditions, the influence of the smallest angle of the triangle on the error of approximation of derivatives of a function by derivatives of the interpolation polynomial is essential for a number of derivatives of order 2 and higher for m ≥ 1. In the present paper, a similar result is proved for some class of composite finite elements.

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References

  1. P. G. Ciarlet and P. A. Raviart, “General Lagrange and Hermite interpolation in R n with applications to finite element methods,” Arch. Rational Mech. Anal. 46(3), 177–199 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Ženišek, “Interpolation polynomials on the triangle,” Numer. Math. 15, 283–296 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  3. J. H. Bramble and M. Zlamal, “Triangular elements in the finite element method,” Math. Comp. 24(112), 809–820 (1970).

    Article  MathSciNet  Google Scholar 

  4. M. Zlamal and A. Ženišek, “Mathematical aspect of the finite element method,” in Technical, Physical and Mathematical Principles of the Finite Element Method, Ed. by V. Kolar et al. (Acad. VED, Prague, 1971), pp. 15–39.

    Google Scholar 

  5. J. L. Synge, The Hypercircle in Mathematical Physics: A Method for the Approximate Solution of Boundary Value Problems (Cambridge Univ. Press, Cambridge, 1957).

    MATH  Google Scholar 

  6. I. Babuška and A. K. Aziz, “On the angle condition in the finite element method,” SIAM J. Numer. Anal. 13(2), 214–226 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  7. Yu. N. Subbotin, “The dependence of estimates of a multidimensional piecewise-polynomial approximation on the geometric characteristics of a triangulation,” Proc. Steklov Inst. Math. 4, 135–159 (1990).

    Google Scholar 

  8. Yu. N. Subbotin, “Dependence of estimates of an approximation by interpolation polynomials of the fifth degree on the geometric characteristics of the triangle,” Trudy Inst. Mat. Mekh. UrO RAN 2, 110–119 (1992).

    MATH  MathSciNet  Google Scholar 

  9. N. V. Latypova, “Error estimates for approximation by polynomials of degree 4k + 3 on the triangle,” Proc. Steklov Inst. Math., Suppl. 1, S190–S213 (2002).

    Google Scholar 

  10. N. V. Baidakova, “On some interpolation process by polynomials of degree 4m + 1 on the triangle,” Russian J. Numer. Anal. Math. Modelling 14(2), 87–107 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  11. Yu. N. Subbotin, “A new cubic element in the FEM,” Proc. Steklov Inst. Math., Suppl. 2, S176–S187 (2005).

    Google Scholar 

  12. N. V. Baidakova, “A method of Hermite interpolation by polynomials of the third degree on a triangle,” Proc. Steklov Inst. Math., Suppl. 2, S49–S55 (2005).

    Google Scholar 

  13. A. Ženišek, “Maximum-angle condition and triangular finite elements of Hermite type,” Math. Comp. 64(211), 929–941 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  14. N. V. Latypova, “Error of piecewise cubic interpolation on a triangle,” Vestn. Udmurt. Univ., Ser. Mat., 3–10 (2003).

    Google Scholar 

  15. Yu. V. Matveeva, “On the Hermite interpolation by polynomials of the third degree on a triangle with the use of mixed derivatives,” Izv. Saratovsk. Univ. 7(1), 23–27 (2007).

    MathSciNet  Google Scholar 

  16. N. V. Baidakova, “Upper estimates for the error of approximation of derivatives in a finite element of Hsieh-Clough-Tocher type,” Trudy Inst. Mat. Mekh. UrO RAN 18(4), 80–89 (2012).

    Google Scholar 

  17. N. V. Baidakova, “Influence of smoothness on the error of approximation of derivatives under local interpolation on triangulations,” Proc. Steklov Inst. Math. 277(Suppl. 1), S33–S47 (2012).

    Article  MATH  Google Scholar 

  18. N. P. Korneichuk, Exact Constants in Approximation Theory (Nauka, Moscow, 1987; Cambridge Univ. Press, Cambridge, 1991).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    N. V. Baidakova

  2. Institute of Mathematics and Computer Science, Ural Federal University, pr. Lenina 51, Yekaterinburg, 620000, Russia

    N. V. Baidakova

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  1. N. V. Baidakova
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Correspondence to N. V. Baidakova.

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Original Russian Text © N.V. Baidakova, 2014, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, Vol. 20, No. 1.

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Baidakova, N.V. Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness property. Proc. Steklov Inst. Math. 288 (Suppl 1), 29–39 (2015). https://doi.org/10.1134/S0081543815020042

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