453 research outputs found
A class of nonparametric DSSY nonconforming quadrilateral elements
A new class of nonparametric nonconforming quadrilateral finite elements is
introduced which has the midpoint continuity and the mean value continuity at
the interfaces of elements simultaneously as the rectangular DSSY element
[J.Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming {G}alerkin
methods based on quadrilateral elements for second order elliptic problems.
ESAIM--Math. Model. Numer. Anal., 33(4):747--770, 1999]. The parametric DSSY
element for general quadrilaterals requires five degrees of freedom to have an
optimal order of convergence [Z. Cai, J. Douglas, Jr., J. E. Santos, D. Sheen,
and X. Ye. Nonconforming quadrilateral finite elements: A correction. Calcolo,
37(4):253--254, 2000], while the new nonparametric DSSY elements require only
four degrees of freedom. The design of new elements is based on the
decomposition of a bilinear transform into a simple bilinear map followed by a
suitable affine map. Numerical results are presented to compare the new
elements with the parametric DSSY element.Comment: 20 page
Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids
In this paper, we first construct a nonconforming finite element pair for the
incompressible Stokes problem on quadrilateral grids, and then construct a
discrete Stokes complex associated with that finite element pair. The finite
element spaces involved consist of piecewise polynomials only, and the
divergence-free condition is imposed in a primal formulation. Combined with
some existing results, these constructions can be generated onto grids that
consist of both triangular and quadrilateral cells
Stable cheapest nonconforming finite elements for the Stokes equations
We introduce two pairs of stable cheapest nonconforming finite element space
pairs to approximate the Stokes equations. One pair has each component of its
velocity field to be approximated by the nonconforming quadrilateral
element while the pressure field is approximated by the piecewise constant
function with globally two-dimensional subspaces removed: one removed space is
due to the integral mean--zero property and the other space consists of global
checker--board patterns. The other pair consists of the velocity space as the
nonconforming quadrilateral element enriched by a globally
one--dimensional macro bubble function space based on
(Douglas-Santos-Sheen-Ye) nonconforming finite element space; the pressure
field is approximated by the piecewise constant function with mean--zero space
eliminated. We show that two element pairs satisfy the discrete inf-sup
condition uniformly. And we investigate the relationship between them. Several
numerical examples are shown to confirm the efficiency and reliability of the
proposed methods
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