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

Richard S. Falk; Paolo Gatto; Peter Monk

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

  • Volume: 45, Issue: 1, page 115-143
  • ISSN: 0764-583X

Abstract

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We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using the appropriate transform (e.g., the Piola transform) associated to a trilinear isomorphism of the cube onto the hexahedron. After determining what vector fields are needed on the reference element to insure O(h) approximation in L2(Ω) and in H(div;Ω) and H(curl;Ω) on the physical element, we study the properties of the resulting finite element spaces.

How to cite

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Falk, Richard S., Gatto, Paolo, and Monk, Peter. "Hexahedral H(div) and H(curl) finite elements." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.1 (2011): 115-143. <http://eudml.org/doc/273342>.

@article{Falk2011,
abstract = {We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using the appropriate transform (e.g., the Piola transform) associated to a trilinear isomorphism of the cube onto the hexahedron. After determining what vector fields are needed on the reference element to insure O(h) approximation in L2(Ω) and in H(div;Ω) and H(curl;Ω) on the physical element, we study the properties of the resulting finite element spaces.},
author = {Falk, Richard S., Gatto, Paolo, Monk, Peter},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {hexahedral finite element; hexahedral finite element spaces; ; &#x03A9;mathvariantupright); ; &#x03A9;mathvariantupright)},
language = {eng},
number = {1},
pages = {115-143},
publisher = {EDP-Sciences},
title = {Hexahedral H(div) and H(curl) finite elements},
url = {http://eudml.org/doc/273342},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Falk, Richard S.
AU - Gatto, Paolo
AU - Monk, Peter
TI - Hexahedral H(div) and H(curl) finite elements
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2011
PB - EDP-Sciences
VL - 45
IS - 1
SP - 115
EP - 143
AB - We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using the appropriate transform (e.g., the Piola transform) associated to a trilinear isomorphism of the cube onto the hexahedron. After determining what vector fields are needed on the reference element to insure O(h) approximation in L2(Ω) and in H(div;Ω) and H(curl;Ω) on the physical element, we study the properties of the resulting finite element spaces.
LA - eng
KW - hexahedral finite element; hexahedral finite element spaces; ; &#x03A9;mathvariantupright); ; &#x03A9;mathvariantupright)
UR - http://eudml.org/doc/273342
ER -

References

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