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Function and Curl Recovery for the Lowest Order Triangular Edge Element

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Abstract

For the lowest order triangular edge element, function and curl recovery methods are proposed to recover the finite element approximation and its curl onto the space of piecewise continuous functions by least-squares fitting. A superconvergence analysis is carried out on the uniform triangular mesh. Numerical experiments are provided to illustrate the superconvergence of the recovery methods and the performance of the corresponding recovery based a posteriori error estimators in adaptive computation.

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References

  1. Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability. Oxford Science Publications, Oxford (2001)

    MATH  Google Scholar 

  2. Bank, R., Xu, J.: Asymptotically exact A posteriori error estimators, part I: grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294–2312 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bramble, J., Schatz, A.: Higher order local accuracy by averaging in the finite element method. Math. Comput. 31, 74–111 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (2008)

    Book  MATH  Google Scholar 

  5. Costabel, M., Dauge, M.: Maxwell and Lamé eigenvalues on polyhedra. Math. Meth. Appl. Sci. 22, 243–258 (1999)

    Article  MATH  Google Scholar 

  6. Chen, C.: Optimal points of the stresses for triangular linear element. Numer. Math. J. Chin. Univ. 2, 12–20 (1980)

    MathSciNet  MATH  Google Scholar 

  7. Chen, C., Huang, Y.: High Accuracy Theory of Finite Element Methods. Hunan Science and Technique Press, Changsha (1995). (in Chinese)

  8. Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (2002)

    Book  MATH  Google Scholar 

  9. Douglas, J., Dupont, T.: Superconvergence for Galerkin methods for the two point boundary problem via local projections. Numer. Math. 21(3), 270–278 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dörfler, W.: A convergent adaptive algorithm for Poisson equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, Y., Li, J., Lin, Q.: Superconvergence analysis for time dependent Maxwell’s equations in metamaterials. Numer. Methods Partial Differ. Equ. 28(6), 1794–1816 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, Y., Li, J., Wu, C., Yang, W.: Superconvergence analysis for linear tetrahedral edge elements. J. Sci. Comput. 62(1), 122–145 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, Y., Li, J., Wu, C.: Averaging for superconvergence: verification and application of 2D edge elements to Maxwell’s equations in metamaterials. Comput. Methods Appl. Mech. Eng. 255, 121–132 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang, Y., Li, J., Wu, C.: Superconvergence analysis of second and third order rectangular edge elements with applications to Maxwell’s equations. Comput. Methods Appl. Mech. Eng. 329, 195–218 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, Y., Li, J., Yang, W., Sun, S.: Superconvergence of mixed finite element approximations to 3-D Maxwell’s equations in metamaterials. J. Comput. Phys. 230(22), 8275–8289 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, Y., Yi, N.: The superconvergent cluster recovery method. J. Sci. Comput. 44, 301–322 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ilic, M., Notaros, B.: Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling. IEEE Trans. Microw. Theory Tech. 51(3), 1026–1033 (2003)

    Article  Google Scholar 

  18. Jin, J.: The Finite Element Method in Electromagnetics. Wiley, New York (2002)

    MATH  Google Scholar 

  19. Li, J., Huang, Y., Yang, W.: An adaptive edge finite element method for electromagnetic cloaking simulation. J. Comput. Phys. 249, 216–232 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lin, Q., Li, J.: Superconvergence analysis for Maxwell’s equations in dispersive media. Math. Comp. 77(262), 757–771 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lin, Q., Yan, N.: Global superconvergence for Maxwell’s equations. Math. Comp. 69(229), 159–176 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lin, Q., Lin, J.: High accuracy approximation of mixed finite element for 2-d Maxwell’s equations. Acta Math. Sci. 23(4), 499–503 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Lin, J., Lin, Q.: Global Superconvergence of the mixed finite element methods for 2D Maxwell equations. J. Comput. Math. 5, 637–646 (2003)

    MATH  Google Scholar 

  24. Monk, P.: A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63(1), 243–261 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Monk, P.: Finite Element Methods for Maxwell Equations. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  26. Nedelec, J.: Mixed finite elements in \(R^{3}\). Numer. Math. 35(3), 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wahlbin, L.: Superconvergence in Galerkin Finite Element Methods. Springer-Verlag, Berlin (1995)

    Book  MATH  Google Scholar 

  28. Wang, L., Zhang, Q., Zhang, Z.: Superconvergence analysis and PPR recovery of arbitrary order edge elements for Maxwell’s equations. J. Sci. Comput. 78(2), 1207–1230 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Webb, J.: Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements. IEEE Trans. Antennas Propagat. 47(8), 1244–1253 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wei H., Huang Y.: FEALPy: finite element analysis library in Python, https://github.com/weihuayi/fealpy, Xiangtan University, (2017)

  31. Wu, C., Huang, Y., Lu, W., Xiong, Z., Yuan, J.: Superconvergence for triangular linear edge elements. Commun. Comput. Phys. 25(4), 1045–1070 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wu, C., Huang, Y., Yi, N., Yuan, J.: Superconvergent recovery of rectangular edge finite element approximation by local symmetry projection. J. Sci. Comput. 81(3), 1602–1629 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wu, C., Huang, Y., Yi, N., Yuan, J.: Superconvergent recovery of edge finite element approximation for Maxwell’s equations. Comput. Methods Appl. Mech. Eng. 371, 113302 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zienkiewicz, O., Zhu, J.: The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique,. Int. J. Numer. Methods Eng. 33(7), 1331–1364 (1992)

  35. Zienkiewicz, O., Zhu, J.: The superconvergent patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity. Int. J. Numer. Methods Eng. 33, 1365–82 (1992)

  36. Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26, 1192–1213 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the referees for their valuable comments which lead to significant improvements in this work. Wu’s research was supported by NSFC project (12226353) and Hunan Provincial NSF Project (2021JJ40189); Huang’s research was partially supported by NSFC project (11971410), China’s National Key R &D Programs (2020YFA0713500) and Hunan National Applied Mathematics Center of Hunan Provincial Science and Technology Department (2020ZYT003); Yi’s research was partially supported by NSFC Project (12071400); Wei’s research was partially supported by the National Natural Science Foundation of China (Grant No. 11871413) and the Construction of Innovative Provinces in Hunan Province (Grant No. 2021GK1010). Yuan’s research was partially supported by NSFC project (12171087).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411100, Hunan, People’s Republic of China

    Chao Wu

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Yunqing Huang, Nianyu Yi & Huayi Wei

  3. School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523808, People’s Republic of China

    Jinyun Yuan

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  1. Chao Wu
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Correspondence to Jinyun Yuan.

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Wu, C., Huang, Y., Yi, N. et al. Function and Curl Recovery for the Lowest Order Triangular Edge Element. J Sci Comput 93, 69 (2022). https://doi.org/10.1007/s10915-022-02027-y

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  • DOI: https://doi.org/10.1007/s10915-022-02027-y

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