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Optimal quadratic element on rectangular grids for \(H^1\) problems

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Abstract

In this paper, a piecewise quadratic finite element method on rectangular grids for \(H^1\) problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is proved to be \(O(h^2)\) in the energy norm on uniform grids over a convex domain. A lower bound of the \(L^2\)-norm error is also proved, which makes the capacity of this scheme more clear. For the eigenvalue problem, the computed eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented to verify the theoretical findings.

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References

  1. Armentano, M.G., Durán, R.G.: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. Electron. Trans. Numer. Anal. 17(2), 93–101 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N., Falk, R.S., Winther, R.: Differential complexes and stability of finite element methods I. the de Rham complex. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds.) Compatible Spatial Discretizations, pp. 23–46. Springer, New York (2006)

  3. Babuška, I., Osborn, J.: Eigenvalue problems. In: Finite Element Methods (Part I), Handbook of Numerical Analysis, vol. 2, pp. 641–787. Elsevier (1991)

  4. Blum, H., Rannacher, R., Leis, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2(4), 556–581 (1980)

    Article  MathSciNet  Google Scholar 

  5. Carstensen, C., Gallistl, D.: Guaranteed lower eigenvalue bounds for the biharmonic equation. Numer. Math. 126(1), 33–51 (2014)

    Article  MathSciNet  Google Scholar 

  6. Carstensen, C., Gedicke, J.: Guaranteed lower bounds for eigenvalues. Math. Comput. 83(290), 2605–2629 (2014)

    Article  MathSciNet  Google Scholar 

  7. Chen, C.: Finite Element Superconvergence Structure Theory. Hunan Science and Technology Press, Hunan (2001)

    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  Google Scholar 

  9. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, vol. 5. Springer, Berlin (2012)

    MATH  Google Scholar 

  10. Hu, J., Huang, Y., Lin, Q.: Lower bounds for eigenvalues of elliptic operators: by nonconforming finite element methods. J. Sci. Comput. 61(1), 196–221 (2014)

    Article  MathSciNet  Google Scholar 

  11. Hu, J., Shi, Z.: The best \(L^{2}\) norm error estimate of lower order finite element methods for the fourth order problem. J. Comput. Math. 30(5), 449–460 (2012)

    Article  MathSciNet  Google Scholar 

  12. Hu, J., Shi, Z.: A lower bound of the \(L^{2}\) norm error estimate for the Adini element of the biharmonic equation. Siam J. Numer. Anal. 51(5), 2651–2659 (2013)

    Article  MathSciNet  Google Scholar 

  13. Hu, J., Yang, X., Zhang, S.: Capacity of the Adini element for biharmonic equations. J. Sci. Comput. 69(3), 1366–1383 (2016)

    Article  MathSciNet  Google Scholar 

  14. Hu, J., Zhang, S.: Nonconforming finite element methods on quadrilateral meshes. Sci. China Math. 56(12), 2599–2614 (2013)

    Article  MathSciNet  Google Scholar 

  15. Hu, J., Zhang, S.: The minimal conforming \(H^k\) finite element spaces on \(\mathbb{R}^n\) rectangular grids. Math. Comput. 84(292), 563–579 (2015)

    Article  Google Scholar 

  16. 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  Google Scholar 

  17. Lee, H., Sheen, D.: A new quadratic nonconforming finite element on rectangles. Numer. Methods Partial Differ. Equ. 22(4), 954–970 (2006)

    Article  MathSciNet  Google Scholar 

  18. Li, Y.: The lower bounds of eigenvalues by the Wilson element in any dimension. Adv. Appl. Math. Mech. 3(5), 598–610 (2011)

    Article  MathSciNet  Google Scholar 

  19. Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. Science Press, Beijing (2006)

    Google Scholar 

  20. Lin, Q., Tobiska, L., Zhou, A.: Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25(1), 160–181 (2005)

    Article  MathSciNet  Google Scholar 

  21. Lin, Q., Xie, H., Xu, J.: Lower bounds of the discretization error for piecewise polynomials. Math. Comput. 83(285), 1–13 (2014)

    Article  MathSciNet  Google Scholar 

  22. Luo, F., Lin, Q., Xie, H.: Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods. Sci. China Math. 55(5), 1069–1082 (2012)

    Article  MathSciNet  Google Scholar 

  23. Meng, X., Yang, X., Zhang, S.: Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions. Sci. China Math. 59(11), 2245–2264 (2016)

    Article  MathSciNet  Google Scholar 

  24. Park, C., Sheen, D.: \(P_{1}\)-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41(2), 624–640 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Shi, Z.: A remark on the optimal order of convergence of Wilson’s nonconforming element. Math. Numer. Sinica 8(2), 159–163 (1986)

    MATH  Google Scholar 

  27. Shi, Z., Wang, M.: Finite Element Methods. Science Press, Beijing (2013)

    Google Scholar 

  28. Wang, M., Xu, J.: Minimal finite element spaces for \(2m\)-th-order partial differential equations in \(\mathbb{R}^n\). Math. Comput. 82(281), 25–43 (2012)

    Article  Google Scholar 

  29. Wilson, E., Taylor, R., Doherty, W., Ghaboussi, J.: Incompatible displacement models. In: Fenves, S.J., Perrone, N., Robinson, A.R., Schnobrich, W.C. (eds.) Numerical and Computer Methods in Structural Mechanics, pp. 43–57. Academic Press, New York (1973)

    Google Scholar 

  30. Wu, S., Xu, J.: Nonconforming finite element spaces for \(2 m\)-th order partial differential equations on \(\mathbb{R}^n\) simplicial grids when \(m= n+ 1\). Math. Comput. 88(316), 531–551 (2019)

    Article  Google Scholar 

  31. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)

    Article  MathSciNet  Google Scholar 

  32. Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70(233), 17–25 (2001)

    Article  MathSciNet  Google Scholar 

  33. Yang, Y., Chen, Z.: The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators. Sci. China Ser. A: Math. 51(7), 1232–1242 (2008)

    Article  MathSciNet  Google Scholar 

  34. Yang, Y., Han, J., Bi, H., Yu, Y.: The lower/upper bound property of the Crouzeix-Raviart element eigenvalues on adaptive meshes. J. Sci. Comput. 62(1), 284–299 (2015)

    Article  MathSciNet  Google Scholar 

  35. Yang, Y., Zhang, Z., Lin, F.: Eigenvalue approximation from below using non-conforming finite elements. Sci. China Ser. A: Math. 53(1), 137–150 (2010)

    Article  MathSciNet  Google Scholar 

  36. Zeng, H., Zhang, C., Zhang, S.: Optimal quadratic element on rectangular grids for \(H^1\) problems (2019). arXiv:1903.00938

  37. Zhang, S.: Stable finite element pair for Stokes problem and discrete stokes complex on quadrilateral grids. Numer. Math. 133(2), 371–408 (2016)

    Article  MathSciNet  Google Scholar 

  38. Zhang, S.: Minimal consistent finite element space for the biharmonic equation on quadrilateral grids. IMA J. Numer. Anal. 40, 1390–1406 (2019). https://doi.org/10.1093/imanum/dry096

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, Z., Yang, Y., Zhen, C.: Eigenvalue approximation from below by Wilson’s element. Math. Numer. Sinica 29(3), 319–321 (2007)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

Zeng and C.-S. Zhang are partially supported by National Key Research and Development Program 2016YFB0201304, China. C.-S. Zhang is also supported by the Key Research Program of Frontier Sciences of CAS. S. Zhang is partially supported by National Natural Science Foundation, 11471026 and 11871465, China.

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Authors and Affiliations

  1. LSEC, ICMSEC and NCMIS, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Huilan Zeng, Chen-Song Zhang & Shuo Zhang

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  1. Huilan Zeng
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  2. Chen-Song Zhang
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  3. Shuo Zhang
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Correspondence to Shuo Zhang.

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Communicated by Axel Målqvist.

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Zeng, H., Zhang, CS. & Zhang, S. Optimal quadratic element on rectangular grids for \(H^1\) problems. Bit Numer Math 61, 665–689 (2021). https://doi.org/10.1007/s10543-020-00821-4

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