HexahedralH(div) andH(curl) finite elements

Abstract

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 L 2 (Ω) and in H(div;Ω) and H(curl;Ω) on the physical element, we study the properties of the resulting finite element spaces.Mathematics Subject Classification. 65N30.

Finite element differential forms on curvilinear cubic meshes and their approximation properties

“…Necessary and sufficient conditions on the space of reference shape functions were given for multilinearly mapped cubical elements in the case of 0-forms in two and three dimensions in [5] and [18], respectively. For the case of (n − 1)-forms, i.e., H (div) finite elements, such conditions were given in n = 2 dimensions in [6], in the lowest order case r = 1 in three dimensions in [16], and for general r in three dimensions in [9]. Necessary and sufficient conditions for 1-forms in three dimensions (H (curl) elements) were also given in the lowest order case in [16], and a closely related problem for H (curl) elements in 3D was studied in [10].…”

“…For the case of (n − 1)-forms, i.e., H (div) finite elements, such conditions were given in n = 2 dimensions in [6], in the lowest order case r = 1 in three dimensions in [16], and for general r in three dimensions in [9]. Necessary and sufficient conditions for 1-forms in three dimensions (H (curl) elements) were also given in the lowest order case in [16], and a closely related problem for H (curl) elements in 3D was studied in [10]. These necessary and sufficient conditions in three dimensions are quite complicated, and would be more so in the general case.…”

“…In that case the transformation of the shape functions must be done through the Piola transform and it turns out the same issue arises, but even more strongly, the requirement on the reference shape functions needed to ensure optimal order approximation being more stringent. Some results were obtained for H (curl) and H (div) finite elements in 3-D in [16].…”

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

“…The needed and sufficient conditions in order to obtain optimal error estimates for H(div) are the following ones (see (10)…”

“…Moreover, these elements will be constructed such that basis functions on the reference elementû will belong to finite element spacesP r that do not depend on the geometry, that is to say they do not depend on the real element K. Such finite elements have been obtained for general quadrilateral elements in (9), and for lowest-order hexahedral elements in (10). In the theorem 3, these super-optimal spaces are detailed at any order for the four types of elements, the lowest order hexahedral space is the same as in (10). However, these spaces are not unique and not very practical to implement, optimal spaces will be detailed in the theorem 2 with more attractive properties.…”