### 2001-03

#### Mapped Finite Elements on Hexahedra. Necessary and Sufficient Conditions for Optimal Interpolation Errors

by Matthies, G..

Series: 2001-03, Preprints

MSC:
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65N15 Error bounds

Abstract:
This paper considers finite elements which are defined on hexahedral cells
via a reference transformation which is in general trilinear. For affine
reference mappings, the necessary and sufficient condition for an
interpolation order ${\cal O}(h^{k+1})$ in the $L^2$-norm and
${\cal O}(h^k)$ in the
$H^1$-norm is that the finite dimensional function space on the reference
cell contains all polynomials of degree less than or equal to $k$. The
situation changes in the case of a general trilinear reference
transformation. We will show that on general meshes the necessary and
sufficient condition for an optimal order for the interpolation error is that
the space of polynomials of degree less than or equal to $k$ in each variable
separately is contained in the function space on the reference cell.
Furthermore, we will show that this condition can be weakened on special
families of meshes. These families which are obtained by applying usual
refinement techniques can be characterized by the asymptotic behaviour of
the semi norms of the reference mapping.

Keywords:
finite element method, hexahedra, interpolation error

This paper was published in:
Numer