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hexahedral finite element

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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Hexahedral H(div) and H(curl) finite elements *
Richard S. Falk, Paolo Gatto, Peter Monk
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 45 / Issue 1 / January 2011

Published online by Cambridge University Press:

10 May 2010, pp. 115-143

Print publication:

January 2011
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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.