2,532 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
h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems
In this work we exploit agglomeration based -multigrid preconditioners to
speed-up the iterative solution of discontinuous Galerkin discretizations of
the Stokes and Navier-Stokes equations. As a distinctive feature -coarsened
mesh sequences are generated by recursive agglomeration of a fine grid,
admitting arbitrarily unstructured grids of complex domains, and agglomeration
based discontinuous Galerkin discretizations are employed to deal with
agglomerated elements of coarse levels. Both the expense of building coarse
grid operators and the performance of the resulting multigrid iteration are
investigated. For the sake of efficiency coarse grid operators are inherited
through element-by-element projections, avoiding the cost of numerical
integration over agglomerated elements. Specific care is devoted to the
projection of viscous terms discretized by means of the BR2 dG method. We
demonstrate that enforcing the correct amount of stabilization on coarse grids
levels is mandatory for achieving uniform convergence with respect to the
number of levels. The numerical solution of steady and unsteady, linear and
non-linear problems is considered tackling challenging 2D test cases and 3D
real life computations on parallel architectures. Significant execution time
gains are documented.Comment: 78 pages, 7 figure
On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study
It is well known that the solution of topology optimization problems may be
affected both by the geometric properties of the computational mesh, which can
steer the minimization process towards local (and non-physical) minima, and by
the accuracy of the method employed to discretize the underlying differential
problem, which may not be able to correctly capture the physics of the problem.
In light of the above remarks, in this paper we consider polygonal meshes and
employ the virtual element method (VEM) to solve two classes of paradigmatic
topology optimization problems, one governed by nearly-incompressible and
compressible linear elasticity and the other by Stokes equations. Several
numerical results show the virtues of our polygonal VEM based approach with
respect to more standard methods
A family of quadrilateral finite elements
We present a novel family of quadrilateral finite elements, which
define global spaces over a general quadrilateral mesh with vertices of
arbitrary valency. The elements extend the construction by (Brenner and Sung,
J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product
degree , to all degrees . Thus, we call the family of
finite elements Brenner-Sung quadrilaterals. The proposed quadrilateral
can be seen as a special case of the Argyris isogeometric element of (Kapl,
Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar
degrees of freedom as the classical Argyris triangles. Just as for the Argyris
triangle, we additionally impose continuity at the vertices. In this
paper we focus on the lower degree cases, that may be desirable for their lower
computational cost and better conditioning of the basis: We consider indeed the
polynomial quadrilateral of (bi-)degree~, and the polynomial degrees
and by employing a splitting into or polynomial
pieces, respectively.
The proposed elements reproduce polynomials of total degree . We show that
the space provides optimal approximation order. Due to the interpolation
properties, the error bounds are local on each element. In addition, we
describe the construction of a simple, local basis and give for
explicit formulas for the B\'{e}zier or B-spline coefficients of the basis
functions. Numerical experiments by solving the biharmonic equation demonstrate
the potential of the proposed quadrilateral finite element for the
numerical analysis of fourth order problems, also indicating that (for )
the proposed element performs comparable or in general even better than the
Argyris triangle with respect to the number of degrees of freedom
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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