by Schieweck, F., Tobiska, L..

**Series:** 2000-18, Preprints

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

**Abstract:**

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.

**Keywords:**

nonconforming quadrilateral finite elements, approximation properties, interpolation error estimates