Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-19T12:20:56.900Z Has data issue: false hasContentIssue false
  • English
  • Français

Hexahedral H(div) and H(curl) finite elements*

Published online by Cambridge University Press:  10 May 2010

Richard S. Falk ,
Paolo Gatto  and
Peter Monk
Richard S. Falk
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA. [email protected]
Paolo Gatto
Affiliation:
Inst. for Comp. Engineering and Sciences, University of Texas at Austin, Austin, TX 78712, USA. [email protected]
Peter Monk
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA. [email protected]
Get access

Abstract

We study the approximation properties of some finite element subspaces ofH(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. Thiswork extends results previously obtained for quadrilateral H(div;Ω) finiteelements and for quadrilateral scalar finite element spaces. The finiteelement spaces we consider are constructed starting from a given finitedimensional space of vector fields on the reference cube, which is thentransformed to a space of vector fields on a hexahedron using the appropriatetransform (e.g., the Piola transform) associated to a trilinear isomorphism ofthe cube onto the hexahedron. After determining what vector fields are neededon the reference element to insure O(h) approximation in L 2(Ω) andin H(div;Ω) and H(curl;Ω) on the physical element, we study the properties ofthe resulting finite element spaces.

Keywords

Hexahedral finite element
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 1 , January 2011 , pp. 115 - 143
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, D.N., Boffi, D., Falk, R.S. and Gastaldi, L., Finite element approximation on quadrilateral meshes. Comm. Num. Meth. Eng. 17 (2001) 805–812. CrossRef
Arnold, D.N., Boffi, D. and Falk, R.S., Approximation by quadrilateral finite elements. Math. Comp. 71 (2002) 909–922. CrossRef
Arnold, D.N., Boffi, D. and Falk, R.S., Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429–2451. CrossRef
I. Babuška and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Proc. Sympos., Univ. Maryland, Baltimore, Md. Academic Press, New York (1972) 1–359.
A. Bermúdez, P. Gamallo, M.R. Nogeiras and R. Rodríguez, Approximation properties of lowest-order hexahedral Raviart-Thomas elements. C. R. Acad. Sci. Paris, Sér. I 340 (2005) 687–692.
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York (1994).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer, New York (1991).
Dubois, F., Discrete vector potential representation of a divergence free vector field in three dimensional domains: Numerical analysis of a model problem. SINUM 27 (1990) 1103–1142. CrossRef
Dupont, T. and Scott, R., Polynomial Approximation of Functions in Sobolev Spaces. Math. Comp. 34 (1980) 441–463. CrossRef
P. Gatto, Elementi finiti su mesh di esaedri distorti per l'approssimazione di H(div) [Approximation of H(div) via finite elements over meshes of distorted hexahedra]. Master's Thesis, Dipartimento di Matematica, Università Pavia, Italy (2006).
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986).
Naff, R.L., Russell, T.F. and Wilson, J.D., Shape Functions for Velocity Interpolation in General Hexahedral Cells. Comput. Geosci. 6 (2002) 285–314. CrossRef
Russell, T.F., Heberton, C.I., Konikow, L.F. and Hornberger, G.Z., A finite-volume ELLAM for three-dimensional solute-transport modeling. Ground Water 41 (2003) 258–272. CrossRef
P. Šolín, K. Segeth and I. Doležel, Higher Order Finite Elements Methods, Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton (2004).
Zhang, S., On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation. Numer. Math. 98 (2004) 559–579. CrossRef
S. Zhang, Invertible Jacobian for hexahedral finite elements. Part 1. Bijectivity. Preprint available at http://www.math.udel.edu/ szhang/research/p/bijective1.ps (2005).
S. Zhang, Invertible Jacobian for hexahedral finite elements. Part 2. Global positivity. Preprint available at http://www.math.udel.edu/ szhang/research/p/bijective2.ps (2005).
S. Zhang, Subtetrahedral test for the positive Jacobian of hexahedral elements. Preprint available at http://www.math.udel.edu/ szhang/research/p/subtettest.pdf (2005).