365 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
The nonconforming virtual element method for eigenvalue problems
We analyse the nonconforming Virtual Element Method (VEM) for the
approximation of elliptic eigenvalue problems. The nonconforming VEM allow to
treat in the same formulation the two- and three-dimensional case.We present
two possible formulations of the discrete problem, derived respectively by the
nonstabilized and stabilized approximation of the L^2-inner product, and we
study the convergence properties of the corresponding discrete eigenvalue
problem. The proposed schemes provide a correct approximation of the spectrum,
in particular we prove optimal-order error estimates for the eigenfunctions and
the usual double order of convergence of the eigenvalues. Finally we show a
large set of numerical tests supporting the theoretical results, including a
comparison with the conforming Virtual Element choice
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
P1-Nonconforming finite elements on triangulations into triangles and quadrilaterals
The P1-nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected Lipschitz domains. An explicit a priori analysis for the combination of the Park–Sheen and the Crouzeix–Raviart nonconforming finite element methods is given for second-order elliptic PDEs with inhomogeneous Dirichlet boundary conditions
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