186
Views
3
CrossRef citations to date
0
Altmetric
Original Articles

Robust nonconforming polynomial finite elements over quadrilaterals

, , &
Pages 758-782 | Received 22 Nov 2019, Accepted 22 May 2020, Published online: 18 Jun 2020

References

  • A. Adini and R.W. Clough. Analysis of plate bending by the finite element method. Report submitted to the National Science Foundation, Grant G7337, 1960.
  • D.N. Arnold, D. Boffi, and R.S. Falk, Approximation by quadrilateral finite elements, Math. Comput.71(239) (2002), pp. 909–922. doi: 10.1090/S0025-5718-02-01439-4
  • D.N. Arnold, D. Boffi, and R.S. Falk, Quadrilateral H(div) finite elements, SIAM J. Numer. Anal. 42 (2005), pp. 2429–2451. doi: 10.1137/S0036142903431924
  • Y. Bao, Z. Meng, and Z. Luo, A C0-nonconforming quadrilateral finite element for the fourth-Order elliptic singular perturbation problem, ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1981–2001. doi: 10.1051/m2an/2018033
  • D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications, Springer, Berlin, Heidelberg, 2013.
  • S. Chen, Y. Zhao, and D. Shi, Non C0 nonconforming elements for elliptic fourth order singular perturbation problem, J. Comput. Math. 23(2) (2005), pp. 185–198.
  • P.G. Ciarlet, The finite element method for elliptic problems, studies in mathematics and its applications, Vol. 4. North-Holland Publishing Company, Amsterdam-New York-Oxford, 1978.
  • J.F. Ciavaldini and J.C. Nédélec, Sur l'élément de Fraeijs de Veubeke et Sander, RAIRO Analyse numérique 8 (1974), pp. 29–45.
  • E. Dubach, R. Luce, and J.M. Thomas, Pseudo-conforming polynomial finite elements on quadrilaterals, Int. J. Comput. Math. 86(10–11) (2009), pp. 1798–1816. doi: 10.1080/00207160902759342
  • T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comput. 34(150) (1980), pp. 441–463. doi: 10.1090/S0025-5718-1980-0559195-7
  • R. Falk and M. Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM J. Numer. Anal. 51(2) (2013), pp. 1308–1326. doi: 10.1137/120888132
  • A. Gillette, K. Hu, and S. Zhang, Nonstandard finite element de Rham complexes on cubical meshes. arXiv preprint arXiv: 1804.04390, 2018.
  • J. Guzman, D. Leykekhman, and M. Neilan, A family of non-conforming elements and the analysis of Nitsche's method for a singular perturbed fourth order problem, Calcolo 49 (2012), pp. 95–125. doi: 10.1007/s10092-011-0047-8
  • J. Guzmán and M. Neilan, A family of nonconforming elements for the Brinkman problem, IMA J. Numer. Anal. 32 (2012), pp. 1484–1508. doi: 10.1093/imanum/drr040
  • J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes, Math. Comput. 83(285) (2014), pp. 15–36. doi: 10.1090/S0025-5718-2013-02753-6
  • V. John, A. Linke, C. Merdon, M. Neilan, and L.G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev. 59(3) (2017), pp. 492–544. doi: 10.1137/15M1047696
  • K.A. Mardal, X.-C Tai, and R. Winther, A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal. 40 (2002), pp. 1605–1631. doi: 10.1137/S0036142901383910
  • L.S.D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aeronaut.19 (1968), pp. 149–169. doi: 10.1017/S0001925900004546
  • M. Neilan and D. Sap, Macro Stokes elements on quadrilaterals, Int. J. Numer. Anal. Model. 15(4–5) (2018), pp. 729–745.
  • T. Nilssen, X.-C. Tai, and R. Winther, A robust nonconforming H2-element, Math. Comput. 70(234) (2001), pp. 489–505. doi: 10.1090/S0025-5718-00-01230-8
  • C. Park and D. Sheen, A quadrilateral Morley element for biharmonic equations, Numer. Math. 124 (2013), pp. 395–413. doi: 10.1007/s00211-013-0517-9
  • L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput. 54(190) (1990), pp. 483–493. doi: 10.1090/S0025-5718-1990-1011446-7
  • F. de Verbeke, A conforming finite element for plate bending, J. Solids Struct. 108 (1968), pp. 4–95.
  • M. Wang, Z. Shi, and J. Xu, A new class of Zienkiewicz-type non-conforming element in any dimensions, Numer. Math. 106(2) (2007), pp. 335–347. doi: 10.1007/s00211-007-0063-4
  • L. Wang, Y. Wu, and X. Xie, Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems, Numer. Methods Partial Differ. Equ., 29(3) (2013), pp. 721–737. doi: 10.1002/num.21723
  • X. Xie, J. Xu, and G. Xue, Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models, J. Comput. Math. 26 (2008), pp. 437–455.
  • X. Xu and S. Zhang, A new divergence-free interpolation operator with applications to the Darcy-Stokes-Brinkman equations, SIAM J. Sci. Comput. 32 (2010), pp. 855–874. doi: 10.1137/090751049
  • S. Zhang, X. Xie, and Y. Chen, Low order nonconforming rectangular finite element methods for Dary-Stokes problem, J. Comput. Math. 27 (2009), pp. 400–424.
  • S. Zhang, Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids, Numer. Math. 133 (2016), pp. 371–408. doi: 10.1007/s00211-015-0749-y
  • X. Zhou, Z. Meng, X. Fan, and Z. Luo, Nonconforming polynomial mixed finite element for the Brinkman problem over quadrilateral meshes, Comput. Math. Appl. 76(4) (2018), pp. 877–892. doi: 10.1016/j.camwa.2018.05.027
  • X. Zhou, Z. Meng, and Z. Luo, New nonconforming finite elements on arbitrary convex quadrilateral meshes, J. Comput. Appl. Math. 296 (2016), pp. 798–814. doi: 10.1016/j.cam.2015.11.004

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.