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A note on the approximation properties of a nonconforming quadrilateral finite element

by Schieweck, F., Tobiska, L..

Series: 2000-18, Preprints

65D05 Interpolation
65N15 Error bounds
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

Recently, Cai, Douglas and Ye have proposed a new
nonconforming point-value oriented quadrilateral finite
element with the property that the integral mean value of
the jump of a finite element function vanishes over each
edge of the grid. For a corresponding nonconforming finite
element discretisation of the Laplacian operator, this
property guarantees an optimal estimate of the consistency
error which does not depend on the variation of the shape
of the quadrilateral mesh cells from the shape of a
parallelogram. This is an advantage in comparison to the
so-called ``parametric'' version of the ``point-value
oriented rotated bilinear'' element introduced by Rannacher
and Turek. However, we prove that for the new proposed
quadrilateral element, the interpolation error is not of
optimal order unless the mesh is ``nearly'' of
parallelogram type.

nonconforming quadrilateral finite elements, approximation properties, interpolation error estimates