Close Search Advanced
Journals
Resources
About Us
Open Access

On the Full C<sub>1</sub>-Q<sub>k</sub> Finite Element Spaces on Rectangles and Cuboids

On the Full C<sub>1</sub>-Q<sub>k</sub> Finite Element Spaces on Rectangles and Cuboids
Preview
Add to basket

Year:    2010

Author:    Shangyou Zhang

Advances in Applied Mathematics and Mechanics, Vol. 2 (2010), Iss. 6 : pp. 701–721

Abstract

We study the extensions of the Bogner-Fox-Schmit element to the whole family of Qk continuously differentiable finite elements on rectangular grids, for all k≥3, in 2D and 3D. We show that the newly defined C1 spaces are maximal in the sense that they contain all C1-Qk functions of piecewise polynomials. We give examples of other extensions of C1-Qk elements. The result is consistent with the Strang's conjecture (restricted to the quadrilateral grids in 2D and 3D). Some numerical results are provided on the family of C1 elements solving the biharmonic equation.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m0993

Advances in Applied Mathematics and Mechanics, Vol. 2 (2010), Iss. 6 : pp. 701–721

Published online:    2010-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    21

Keywords:   

Author Details

Shangyou Zhang

  1. Robust Falk-Neilan Finite Elements for the Reissner-Mindlin Plate

    Zhang, Shangyou

    Communications on Applied Mathematics and Computation, Vol. 5 (2023), Iss. 4 P.1697

    https://doi.org/10.1007/s42967-023-00266-w [Citations: 3]

  2. Residual error estimation for anisotropic Kirchhoff plates

    Weise, Michael

    Applied Numerical Mathematics, Vol. 125 (2018), Iss. P.10

    https://doi.org/10.1016/j.apnum.2017.10.007 [Citations: 3]

  3. Nonconforming virtual element method for 2mth order partial differential equations in $${\mathbb {R}}^n$$ with $$m>n$$

    Huang, Xuehai

    Calcolo, Vol. 57 (2020), Iss. 4

    https://doi.org/10.1007/s10092-020-00381-7 [Citations: 12]

  4. On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation

    Cui, Ming | Zhang, Shangyou

    Journal of Scientific Computing, Vol. 82 (2020), Iss. 1

    https://doi.org/10.1007/s10915-019-01120-z [Citations: 27]

  5. $$C^1$$-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

    Li, Huiyuan | Shan, Weikun | Zhang, Zhimin

    Communications on Applied Mathematics and Computation, Vol. 1 (2019), Iss. 3 P.403

    https://doi.org/10.1007/s42967-019-00041-w [Citations: 6]

  6. A family of 3D H2-nonconforming tetrahedral finite elements for the biharmonic equation

    Hu, Jun | Tian, Shudan | Zhang, Shangyou

    Science China Mathematics, Vol. 63 (2020), Iss. 8 P.1505

    https://doi.org/10.1007/s11425-019-1661-8 [Citations: 12]

  7. Conforming and nonconforming finite element methods for canonical von Kármán equations

    Mallik, Gouranga | Nataraj, Neela

    International Journal of Advances in Engineering Sciences and Applied Mathematics, Vol. 7 (2015), Iss. 3 P.86

    https://doi.org/10.1007/s12572-015-0137-y [Citations: 1]

  8. A divergence-free finite element method for a type of 3D Maxwell equations

    Huang, Jianguo | Zhang, Shangyou

    Applied Numerical Mathematics, Vol. 62 (2012), Iss. 6 P.802

    https://doi.org/10.1016/j.apnum.2011.06.009 [Citations: 20]

  9. Reconstructed Interpolating Differential Operator Method with Arbitrary Order of Accuracy for the Hyperbolic Equation

    Lin, Shijian | Luo, Qi | Leng, Hongze | Song, Junqiang

    Axioms, Vol. 10 (2021), Iss. 4 P.295

    https://doi.org/10.3390/axioms10040295 [Citations: 0]

  10. Cut Bogner-Fox-Schmit elements for plates

    Burman, Erik | Hansbo, Peter | Larson, Mats G.

    Advanced Modeling and Simulation in Engineering Sciences, Vol. 7 (2020), Iss. 1

    https://doi.org/10.1186/s40323-020-00164-3 [Citations: 12]

  11. High-precision computation of the weak Galerkin methods for the fourth-order problem

    Burkardt, John | Gunzburger, Max | Zhao, Wenju

    Numerical Algorithms, Vol. 84 (2020), Iss. 1 P.181

    https://doi.org/10.1007/s11075-019-00751-5 [Citations: 9]

  12. A Combination of a Special Hermite Finite Element with Collocation for a Reaction-Diffusion Type Equation

    Gileva, L. V. | Karepova, E. D. | Shaidurov, V. V.

    Lobachevskii Journal of Mathematics, Vol. 40 (2019), Iss. 4 P.459

    https://doi.org/10.1134/S1995080219040085 [Citations: 1]

  13. Accurate numerical integration for the quadrilateral and hexahedral finite elements

    Xie, Ziqing | Zhang, Shangyou

    Numerical Methods for Partial Differential Equations, Vol. 37 (2021), Iss. 3 P.2446

    https://doi.org/10.1002/num.22723 [Citations: 1]
  14. Finite Difference Methods,Theory and Applications

    Superconvergence of Some Linear and Quadratic Functionals for Higher-Order Finite Elements

    Shaydurov, Vladimir | Xu, Tianshi

    2015

    https://doi.org/10.1007/978-3-319-20239-6_8 [Citations: 0]

  15. A $$C^0$$ C 0 -Weak Galerkin Finite Element Method for the Biharmonic Equation

    Mu, Lin | Wang, Junping | Ye, Xiu | Zhang, Shangyou

    Journal of Scientific Computing, Vol. 59 (2014), Iss. 2 P.473

    https://doi.org/10.1007/s10915-013-9770-4 [Citations: 59]

  16. Conforming finite element methods for the von Kármán equations

    Mallik, Gouranga | Nataraj, Neela

    Advances in Computational Mathematics, Vol. 42 (2016), Iss. 5 P.1031

    https://doi.org/10.1007/s10444-016-9452-5 [Citations: 17]
  17. Numerical Mathematics and Advanced Applications ENUMATH 2019

    Augmented Lagrangian Method for Thin Plates with Signorini Boundaries

    Burman, Erik | Hansbo, Peter | Larson, Mats G.

    2021

    https://doi.org/10.1007/978-3-030-55874-1_50 [Citations: 1]

  18. Hierarchical high‐order conforming C1 bases for quadrangular and triangular finite elements

    Ferreira, Luan J. F. | Bittencourt, Marco L.

    International Journal for Numerical Methods in Engineering, Vol. 109 (2017), Iss. 7 P.936

    https://doi.org/10.1002/nme.5308 [Citations: 8]

  19. k-Version of Finite Element Method for BVPs and IVPs

    Surana, Karan S. | Carranza, Celso H. | Mathi, Sri Sai Charan

    Mathematics, Vol. 9 (2021), Iss. 12 P.1333

    https://doi.org/10.3390/math9121333 [Citations: 1]

  20. Locking-free Argyris–Lagrange finite elements for the Reissner–Mindlin plate

    Huang, Yunqing | Zhang, Shangyou

    Calcolo, Vol. 61 (2024), Iss. 3

    https://doi.org/10.1007/s10092-024-00608-x [Citations: 0]

  21. The application of special Hermite finite elements coupled with collocation to the 3D Poisson equation

    Gileva, L. | Karepova, E. | Shaidurov, V.

    APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 11th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS’19, (2019), P.110004

    https://doi.org/10.1063/1.5130849 [Citations: 0]

  22. A new P1 weak Galerkin method for the Biharmonic equation

    Ye, Xiu | Zhang, Shangyou | Zhang, Zhimin

    Journal of Computational and Applied Mathematics, Vol. 364 (2020), Iss. P.112337

    https://doi.org/10.1016/j.cam.2019.07.002 [Citations: 13]

  23. A Locking-Free and Reduction-Free Conforming Finite Element Method for the Reissner-Mindlin Plate on Rectangular Meshes

    Zhang, Shangyou | Zhang, Zhimin

    Communications on Applied Mathematics and Computation, Vol. (2024), Iss.

    https://doi.org/10.1007/s42967-023-00343-0 [Citations: 1]

  24. A Combined Finite-Element-Finite-Volume Method for Thermal Arc Numerical Simulation

    Abdo, Youssef | Rohani, Vandad | Cauneau, François | Fulcheri, Laurent

    Journal of Physics: Conference Series, Vol. 1243 (2019), Iss. 1 P.012011

    https://doi.org/10.1088/1742-6596/1243/1/012011 [Citations: 2]

  25. Hermitian finite element complementing the Bogner–Fox–Schmit rectangle near curvilinear boundary

    Dobronets, B. | Shaydurov, V.

    Lobachevskii Journal of Mathematics, Vol. 37 (2016), Iss. 5 P.527

    https://doi.org/10.1134/S1995080216050036 [Citations: 1]

  26. Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

    Huang, Yunqing | Zhang, Shangyou

    Communications in Mathematics and Statistics, Vol. 1 (2013), Iss. 2 P.143

    https://doi.org/10.1007/s40304-013-0012-8 [Citations: 19]