Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-19T11:06:35.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra

Published online by Cambridge University Press:  03 June 2015

Morgane Bergot  and
Marc Duruflé
Morgane Bergot*
Affiliation:
CALVI project team, INRIA Nancy-Grand Est, Strasbourg, France
Marc Duruflé*
Affiliation:
BACCHUS project team, INRIA Bordeaux Sud-Ouest, Bordeaux, France
*
Corresponding author.Email:marc.durufle@inria.fr
Email:morgane.bergot@inria.fr
Get access

Abstract

Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in H(div-norm for general unstructured meshes containing hexahedra and prisms. We propose two new families of high-order elements for hexahedra, triangular prisms and pyramids that recover the optimal convergence. These elements have compatible restrictions with each other, such that they can be used directly on general hybrid meshes. Moreover the H(div) proposed spaces are completing the De Rham diagram with optimal elements previously constructed for H1 and H(curl) approximation. The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature. Eventually, numerical results demonstrate the efficiency of the finite elements constructed.

Keywords

65M60Facet elementshigh-order finite elementpyramidsH(div) approximationDe Rham diagram
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 5 , November 2013 , pp. 1372 - 1414
Copyright
Copyright © Global Science Press Limited 2013

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

[1]Bergot, M., Cohen, G., Durufleé, M., Higher-order finite elements for hybrid meshes using new nodal pyramidal elements, J. Sci. Comput. 42 (3) (2010) 345–381.Google Scholar
[2]Bergot, M., Durufleé, M., High-order optimal edge elements for pyramids, prisms and hexa-hedra, J. Comp. Phys., accepted.Google Scholar
[3]Cockburn, B., Gopalakrishnan, J., Incompressible finite elements via hybridization.part ii: The stokes system in three space dimensions, SIAM Journal on Numerical Analysis 43 (2005) 1651–1672.Google Scholar
[4]Sboui, A., Jaffreé, J., Roberts, J., A composite mixed finite element for hexahedral grids, SIAM J. Sci. Comput. (2009) 2623–2645.Google Scholar
[5]Nédélec, J. C., Mixed finite elements in R3, Numer. Math. 35 (3) (1980) 315–341.Google Scholar
[6]Nédélec, J. C., A new family of mixed finite elements in R3, Numer. Math. 51 (1) (1986) 57–81.Google Scholar
[7]Raviart, P., Thomas, J., A mixed finite element method for 2nd order elliptic problem, Lecture Notes in Mathematics 606 (1977) 292–315.CrossRefGoogle Scholar
[8]Naff, R. L., Russell, T. F., Wilson, J. D., Shape functions for velocity interpolation in general hexahedral cells, Comput. Geosci. 6 (2002) 285–314.Google Scholar
[9]Arnold, D. N., Boffi, D., Falk, R. S., Quadrilateral H(div) finite elements, SIAM J. Numer. Anal. 42 (6) (2005) 2429–2451.Google Scholar
[10]Falk, R., Gatto, P., Monk, P., Hexahedral H(div) and H(curl) finite elements, ESAIM: M2AN 45 (1) (2011) 115–143.Google Scholar
[11]Owen, S., Saigal, S., Formation of pyramid elements for hexahedra to tetrahedra transitions, Comp. Meth. Appl. Mech. Eng. 190 (2001) 4505–4518.Google Scholar
[12]Nigam, N., Phillips, J., High-order finite elements on pyramids: approximation spaces, unisolvency and exactness., IMA J. of Nu. Anal. 32 (2) (2012) 448–483.Google Scholar
[13]Nigam, N., Phillips, J., Numerical integration for high order pyramidal finite elements, in revision.Google Scholar
[14]Graglia, R. D., Wilton, D. R., Peterson, A. F., Gheorma, I.-L., Higher order interpolatory vector bases on pyramidal elements, IEEE Trans. Ant. Prop. 47 (5) (1999) 775–782.Google Scholar
[15]Monk, P., Finite element methods for Maxwell’s equations, Oxford Science Publication, 2002.Google Scholar
[16]Demkowicz, L., Kurtz, J., Pardo, D., Paszynski, M., Rachowicz, W., Zdunek, A., Computing With hp-Adaptive Finite Elements, Volume II, Chapman & Hall/CRC, 2007.Google Scholar
[17]Dular, P., Hody, J.-Y., Nicolet, A., Genon, A., Legros, W., Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms, IEEE Trans. Mag. 30 (5) (1994) 2980–2983.Google Scholar
[18]Šolín, P., Segeth, K., A new sequence of hierarchic prismatic elements satisfying de rham diagram on hybrid meshes, J. Numer. Math. 13 (2005) 295–318.Google Scholar
[19]Gradinaru, V., Hiptmair, R., Whitney elements on pyramids, Elec. Trans. Num. Anal. 8 (1999) 154–168.Google Scholar