01-03
Mapped Finite Elements on Hexahedra. Necessary and Sufficient Conditions for Optimal Interpolation Errors
by Matthies, G.
Preprint series: 01-03, Preprints
The paper is published: Numer. Algorithm, 27 (2001), 317-327.
- 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
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