Abstract
We study the properties of the reference mapping for quadrilateral and hexahedral finite elements. We consider multilevel adaptive grids with possibly hanging nodes which are typically generated by adaptive refinement starting from a regular coarse grid. It turns out that for such grids the reference mapping behaves – up to a perturbation depending on the mesh size – like an affine mapping. As an application, we prove optimal estimates of the interpolation error for discontinuous mapped \(\mathbb{P}_r\) -elements on quadrilateral and hexahedral grids.
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D. N. Arnold D. Boffi R. S. Falk (2002) ArticleTitleApproximation by quadrilateral finite elements Math. Comput. 71 909–922 Occurrence Handle0993.65125 Occurrence Handle10.1090/S0025-5718-02-01439-4 Occurrence Handle1898739
P. G. Ciarlet (1978) The finite element method for elliptic problems. Studies in Mathematics and its Applications, Vol. 4 North-Holland Amsterdam New York Oxford
H. Federer (1969) Geometric measure theory Springer New York Occurrence Handle0176.00801
V. Heuveline R. Rannacher (2001) ArticleTitleA posteriori error control for finite element approximations of elliptic eigenvalue problems Adv. Comput. Math. 15 107–138 Occurrence Handle0995.65111 Occurrence Handle10.1023/A:1014291224961 Occurrence Handle1887731
V. Heuveline R. Rannacher (2003) ArticleTitleDuality-based adaptivity in the hp-finite element method J. Numer. Math. 11 95–113 Occurrence Handle1050.65111 Occurrence Handle10.1515/156939503766614126 Occurrence Handle1987590
G. Matthies (2001) ArticleTitleMapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors Numer. Algorithm 27 317–327 Occurrence Handle0998.65117 Occurrence Handle10.1023/A:1013860707381 Occurrence Handle1880917
Matthies, G.: Finite element methods for free boundary value problems with capillary surfaces. PhD thesis, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, 2002. Aachen: Shaker-Verlag.
G. Matthies L. Tobiska (2002) ArticleTitleThe inf-sup condition for the mapped Q k -P k -1 disc element in arbitrary space dimensions Computing 69 119–139 Occurrence Handle1016.65073 Occurrence Handle10.1007/s00607-002-1451-3 Occurrence Handle1954791
R. Rannacher S. Turek (1992) ArticleTitleSimple nonconforming quadrilateral stokes element Numer. Meth. Partial Differ. Equ. 8 97–111 Occurrence Handle0742.76051 Occurrence Handle10.1002/num.1690080202 Occurrence Handle1148797
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Matthies, G., Schieweck, F. On the reference mapping for quadrilateral and hexahedral finite elements on multilevel adaptive grids. Computing 80, 95–119 (2007). https://doi.org/10.1007/s00607-007-0228-0
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DOI: https://doi.org/10.1007/s00607-007-0228-0