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

¹ Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA. falk@math.rutgers.edu
² Inst. for Comp. Engineering and Sciences, University of Texas at Austin, Austin, TX 78712, USA. gatto@ices.utexas.edu
³ Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA. monk@math.udel.edu

Received: 27 March 2009
Revised: 18 November 2009

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.