By Richard Falk.
Abstract: We consider the approximation properties of finite element subspaces of H1 and H(div) elements defined on quadrilateral meshes in two dimensions and of H1, H(div), and H(curl) elements defined on hexahedral meshes in three dimensions. The finite element spaces considered are constructed starting from a given finite dimensional space of functions or vector fields on the reference square or cube, which is then transformed to a space of functions or vector fields on a quadrilateral or hexahedron using the appropriate transformation. For three dimensional H(div) elements, this will be the Piola transform associated to a trilinear isomorphism of the cube onto the hexahedron. A main goal is to determine what vector fields are needed on the reference element to insure optimal order approximation in L2, and in H(div) and H(curl) on the physical element, and to construct finite element spaces that have this property. As applications, we show that some commonly used finite element spaces that have optimal approximation properties on rectangular meshes, are suboptimal on quadrilateral or hexahedral meshes.