Skip to main content Accessibility help
×
Home

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

  • Morgane Bergot (a1) and Marc Duruflé (a2)

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.

Copyright

Corresponding author

Corresponding author.Email:marc.durufle@inria.fr

References

Hide All
[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) 345381.
[2]Bergot, M., Durufleé, M., High-order optimal edge elements for pyramids, prisms and hexa-hedra, J. Comp. Phys., accepted.
[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) 16511672.
[4]Sboui, A., Jaffreé, J., Roberts, J., A composite mixed finite element for hexahedral grids, SIAM J. Sci. Comput. (2009) 26232645.
[5]Nédélec, J. C., Mixed finite elements in R3, Numer. Math. 35 (3) (1980) 315341.
[6]Nédélec, J. C., A new family of mixed finite elements in R3, Numer. Math. 51 (1) (1986) 5781.
[7]Raviart, P., Thomas, J., A mixed finite element method for 2nd order elliptic problem, Lecture Notes in Mathematics 606 (1977) 292315.
[8]Naff, R. L., Russell, T. F., Wilson, J. D., Shape functions for velocity interpolation in general hexahedral cells, Comput. Geosci. 6 (2002) 285314.
[9]Arnold, D. N., Boffi, D., Falk, R. S., Quadrilateral H(div) finite elements, SIAM J. Numer. Anal. 42 (6) (2005) 24292451.
[10]Falk, R., Gatto, P., Monk, P., Hexahedral H(div) and H(curl) finite elements, ESAIM: M2AN 45 (1) (2011) 115143.
[11]Owen, S., Saigal, S., Formation of pyramid elements for hexahedra to tetrahedra transitions, Comp. Meth. Appl. Mech. Eng. 190 (2001) 45054518.
[12]Nigam, N., Phillips, J., High-order finite elements on pyramids: approximation spaces, unisolvency and exactness., IMA J. of Nu. Anal. 32 (2) (2012) 448483.
[13]Nigam, N., Phillips, J., Numerical integration for high order pyramidal finite elements, in revision.
[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) 775782.
[15]Monk, P., Finite element methods for Maxwell’s equations, Oxford Science Publication, 2002.
[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.
[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) 29802983.
[18]Šolín, P., Segeth, K., A new sequence of hierarchic prismatic elements satisfying de rham diagram on hybrid meshes, J. Numer. Math. 13 (2005) 295318.
[19]Gradinaru, V., Hiptmair, R., Whitney elements on pyramids, Elec. Trans. Num. Anal. 8 (1999) 154168.

Keywords

Related content

Powered by UNSILO

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

  • Morgane Bergot (a1) and Marc Duruflé (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.