Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
DE
EN
Books
Journals
Events
Topic Page
Marketing
Log in
Learn more about individual access
Request company access
12th International Engine Congress 2025 | 25. + 26. February 2025

Meeting Place for the Powertrain and Sustainable Fuels Community

The congress will examine the future role of the internal combustion engine and its non-fossil fuels from a global perspective with leading experts. Be part of it! More information and booking options are available on our website.
EXTENDED SEARCH
Log in
Top
Foundations of Computational Mathematics
Published in:

17-10-2023

A Construction of \(C^r\) Conforming Finite Element Spaces in Any Dimension

Authors: Jun Hu, Ting Lin, Qingyu Wu

Published in: Foundations of Computational Mathematics | Issue 6/2024

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This paper proposes a construction of \(C^r\) conforming finite element spaces with arbitrary r in any dimension. It is shown that if \(k \ge 2^{d}r+1\) the space \({\mathcal {P}}_k\) of polynomials of degree \(\le k\) can be taken as the shape function space of \(C^r\) finite element spaces in d dimensions. This is the first work on constructing such \(C^r\) conforming finite elements in any dimension in a unified way.

previous article Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Impedance Boundary Conditions
next article Convex Analysis on Hadamard Spaces and Scaling Problems

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Literature
  1. Peter Alfeld, Larry L Schumaker, and Maritza Sirvent, On dimension and existence of local bases for multivariate spline spaces, Journal of Approximation Theory 70 (1992), 243–264.MathSciNetView Article
  2. Peter Alfeld and Maritza Sirvent, The structure of multivariate superspline spaces of high degree, Mathematics of Computation 57 (1991), 299–308.MathSciNetView Article
  3. Paola F Antonietti, G Manzini, and Marco Verani, The conforming virtual element method for polyharmonic problems, Computers & Mathematics with Applications 79 (2020),  2021–2034.MathSciNetView Article
  4. John H Argyris, Isaac Fried, and Dieter W Scharpf, The TUBA family of plate elements for the matrix displacement method, The Aeronautical Journal 72 (1968), 701–709.
  5. L Beirão da Veiga, Franco Brezzi, Andrea Cangiani, Gianmarco Manzini, L Donatella Marini, and Alessandro Russo, Basic principles of virtual element methods, Mathematical Models and Methods in Applied Sciences 23 (2013),  199–214.MathSciNetView Article
  6. L Beirão da Veiga, Franco Brezzi, Luisa Donatella Marini, and Alessandro Russo, The hitchhiker’s guide to the virtual element method, Mathematical Models and Methods in Applied Sciences 24 (2014),  1541–1573.MathSciNetView Article
  7. James H Bramble and Miloš Zlámal, Triangular elements in the finite element method, Mathematics of Computation 24 (1970), 809–820.MathSciNetView Article
  8. Susanne C Brenner and L Ridgway Scott, The mathematical theory of finite element methods, Springer, UK. 2008.View Article
  9. Long Chen and Xuehai Huang, Nonconforming virtual element method for \(2m\) th order partial differential equations in \(\mathbb{R} ^n\), Mathematics of Computation 89 (2020),  1711–1744.MathSciNetView Article
  10. Snorre H Christiansen, Jun Hu, and Kaibo Hu, Nodal finite element de rham complexes, Numerische Mathematik 139 (2018), 411–446.
  11. Charles K Chui and Ming-Jun Lai, Multivariate vertex splines and finite elements, Journal of Approximation Theory 60 (1990), 245–343.MathSciNetView Article
  12. Philippe G Ciarlet, The finite element method for elliptic problems, SIAM, 2002.
  13. Richard S Falk and Michael Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM Journal on Numerical Analysis 51 (2013), 1308–1326.MathSciNetView Article
  14. Jun Hu, Yunqing Huang, and Shangyou Zhang, The lowest order differentiable finite element on rectangular grids, SIAM Journal on Numerical Analysis 49 (2011),  1350–1368.MathSciNetView Article
  15. Jun Hu and Yizhou Liang, Conforming discrete gradgrad-complexes in three dimensions, Mathematics of Computation 90 (2021), 1637–1662.MathSciNetView Article
  16. Jun Hu and Shangyou Zhang, A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids, Science China Mathematics 58 (2015), 297–307.MathSciNetView Article
  17. Jun Hu and Shangyou Zhang, The minimal conforming \(H^k\) finite element spaces on \(\mathbb{R} ^n\) rectangular grids, Mathematics of Computation 84 (2015),  563–579.MathSciNet
  18. Jun Hu and Shangyou Zhang, A canonical construction of hm-nonconforming triangular finite elements, Annals of Applied Mathematics 33 (2017), 266–288.MathSciNet
  19. Xuehai Huang, Nonconforming virtual element method for \(2m\) th order partial differential equations in \(\mathbb{R} ^n\) with \(m>n\), Calcolo 57 (2020),  1–38.MathSciNetView Article
  20. Ming-Jun Lai and Larry L Schumaker, Spline functions on triangulations, Cambridge University Press, 2007.View Article
  21. Leslie Sydney Dennis Morley, The triangular equilibrium element in the solution of plate bending problems, Aeronautical Quarterly 19 (1968), 149–169.View Article
  22. Rolf Stenberg, Error analysis of some finite element methods for the stokes problem, Mathematics of Computation 54 (1990), 495–508.MathSciNetView Article
  23. Ming Wang and Jinchao Xu, The morley element for fourth order elliptic equations in any dimensions, Numerische Mathematik 103 (2006), 155–169.MathSciNetView Article
  24. Ming Wang and Jinchao Xu, Minimal finite element spaces for \(2m\)-th-order partial differential equations in \(\mathbb{R} ^n\), Mathematics of Computation 82 (2013),  25–43.MathSciNetView Article
  25. Shuonan Wu and Jinchao Xu, \(\cal{P}_m\) interior penalty nonconforming finite element methods for \(2m\)-th order PDEs in \(\mathbb{R}^n\), arXiv preprint (2017).
  26. Shuonan Wu and Jinchao Xu, Nonconforming finite element spaces for \(2m\) th order partial differential equations on \(\mathbb{R} ^n\) simplicial grids when \(m=n+1\), Mathematics of Computation 88 (2019),  531–551.MathSciNet
  27. Jinchao Xu, Finite neuron method and convergence analysis, Communications in Computational Physics 28 (2020), 1707–1745.MathSciNetView Article
  28. Alexander Ženíšek, Interpolation polynomials on the triangle, Numerische Mathematik 15 (1970),  283–296.MathSciNetView Article
  29. Shangyou Zhang, A family of 3d continuously differentiable finite elements on tetrahedral grids, Applied Numerical Mathematics 59 (2009), 219–233.MathSciNetView Article
  30. Shangyou Zhang, A family of differentiable finite elements on simplicial grids in four space dimensions, Mathematica Numerica Sinica 38 (2016), 309.MathSciNet
  31. Shangyou Zhang, The nodal basis of \( C^m \)-\( P_k^{(3)}\) and \( C^m \)-\( P_k^{(4)}\) finite elements on tetrahedral and 4d simplicial grids, arXiv preprint (2022).
Metadata
Title
A Construction of Conforming Finite Element Spaces in Any Dimension
Authors
Jun Hu
Ting Lin
Qingyu Wu
Publication date
17-10-2023
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 6/2024
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-023-09627-6

Other articles of this Issue 6/2024

On the Representation and Learning of Monotone Triangular Transport Maps

Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms

Tribute to Nick Higham

  • In Memoriam

From Kernel Methods to Neural Networks: A Unifying Variational Formulation

  • Open Access

Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Impedance Boundary Conditions

  • Open Access

Approximation of Deterministic Mean Field Games with Control-Affine Dynamics

Premium Partner

    Image Credits