A note on the approximation properties of a nonconforming quadrilateral finite element
Preprint series: 00-18, Preprints
- 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
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