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A New Rotated Nonconforming Quadrilateral Element

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Abstract

In this paper, a new nonparametric nonconforming quadrilateral finite element is introduced. This element takes the four edge mean values as the degrees of the freedom and the finite element space is a subspace of \(P_{2}\). Different from the other nonparametric elements, the basis functions of this new element can be expressed explicitly without solving linear systems locally, which can be achieved by introducing a new reference quadrilateral. To evaluate the integration, a class of new quadrature formulae with only three equally weighted points on quadrilateral are constructed. Hence the stiffness matrix can be calculated by the same way with the parametric elements. Numerical results are shown to confirm the optimality of the convergence order for the second order elliptic problems and the Stokes problem.

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References

  1. Arnold, D.N., Winther, R.: Nonconforming mixed elements for elasticity. Math. Models Methods Appl. Sci. 13(3), 295–307 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brenner, S., Sung, L.: Linear finite element methods for planar linear elasticity. Math. Comput. 59(200), 321–338 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, Z., Douglas, J., Ye, X.: A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations. Calcolo 36, 215–232 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, Z., Douglas Jr., J., Santos, J., Sheen, D., Ye, X.: Nonconforming quadrilateral finite elements: a correction. Calcolo 37(4), 253–254 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Crouzeix, M., Raviart, P.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. Rev. Fr. Autom. Inf. Rech. Opér. Math. 7, 33–75 (1973)

    MathSciNet  MATH  Google Scholar 

  6. Douglas Jr., J., Santos, J., Sheen, D., Ye, X.: Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. Math. Model. Numer. Anal. 33(4), 747–770 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fortin, M.: A three-dimensional quadratic nonconforming element. Numer. Math. 46(2), 269–279 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fortin, M., Soulie, M.: A non-conforming piecewise quadratic finite element on triangles. Int. J. Numer. Methods Eng. 19(4), 505–520 (1983)

    Article  MATH  Google Scholar 

  9. Han, H.D.: Nonconforming elements in the mixed finite element method. J. Comput. Math. 2(3), 223–233 (1984)

    MathSciNet  MATH  Google Scholar 

  10. Hu, J., Shi, Z.: Constrained quadrilateral nonconforming rotated \(Q_1\)-element. J. Comput. Math. 23, 561–586 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Jeon, Y., Nam, H., Sheen, D.: A nonconforming quadrilateral element with maximal inf-sup constant. Numer. Methods Partial Differ. Equ. 30(1), 120–132 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jeon, Y., Nam, H., Sheen, D., Shim, K.: A class of nonparametric DSSY nonconforming quadrilateral elements. ESAIM Math. Model. Numer. Anal. 47(6), 1783–1796 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kim, I., Luo, Z., Meng, Z., Nam, H., Park, C., Sheen, D.: A piecewise \(p_2\)-nonconforming quadrilateral finite element. ESAIM Math. Model. Numer. Anal. 47(3), 689–715 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kloucek, P., Li, B., Luskin, M.: Analysis of a class of nonconforming finite elements for crystalline microstructures. Math. Comput. 65(215), 1111–1135 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Köster, M., Ouazzi, A., Schieweck, F., Turek, S., Zajac, P.: New robust nonconforming finite elements of higher order. Appl. Numer. Math. 62(3), 166–184 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lee, C., Lee, J., Sheen, D.: A locking-free nonconforming finite element method for planar linear elasticity. Adv. Comput. Math. 19(1), 277–291 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ming, P., Shi, Z.: Nonconforming rotated \(Q_1\) element for Reissner–Mindlin plate. Math. Models Methods Appl. Sci. 11(8), 1311–1342 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Park, C., Sheen, D.: \(P_1\)-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41(2), 624–640 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8(2), 97–111 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall Inc, Englewood Cliffs (1971)

    MATH  Google Scholar 

  21. Zhou, X., Meng, Z., Luo, Z.: New nonconforming finite elements on arbitrary convex quadrilateral meshes. J. Comput. Appl. Math. 296, 798–814 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Zhaoliang Meng & Zhongxuan Luo

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hum, Hong Kong, China

    Jintao Cui

  3. School of Software, Dalian University of Technology, Dalian, 116620, China

    Zhongxuan Luo

Authors
  1. Zhaoliang Meng
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  2. Jintao Cui
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  3. Zhongxuan Luo
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Corresponding author

Correspondence to Zhaoliang Meng.

Additional information

This Project is supported by NNSFC (Nos.11301053, 61432003).

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Meng, Z., Cui, J. & Luo, Z. A New Rotated Nonconforming Quadrilateral Element. J Sci Comput 74, 324–335 (2018). https://doi.org/10.1007/s10915-017-0435-6

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  • DOI: https://doi.org/10.1007/s10915-017-0435-6

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