共查询到20条相似文献,搜索用时 15 毫秒
1.
Crouzeix-Raviart type finite elements on anisotropic meshes 总被引:47,自引:0,他引:47
Summary. The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element
is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is
proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from
classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity
assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate,
namely the Poisson problem in domains with edges. A numerical test is described.
Received May 19, 1999 / Revised version received February 2, 2000 / Published online February 5, 2001 相似文献
2.
Summary.
We consider convex interpolation with
cubic splines on
grids built by adding two knots in each subinterval of neighbouring data
sites. The additional knots have to be variable in order to get a chance
to always retain convexity. By means of the staircase algorithm we
provide computable intervals for the added knots such that all knots
from these intervals allow convexity preserving spline interpolation of
continuity.
Received
May 31, 1994 / Revised version received December 22, 1994 相似文献
3.
Yuan Xu 《Numerische Mathematik》1994,69(2):233-241
Summary.
The existence of Gaussian cubature for a given measure
depends on whether the corresponding multivariate orthogonal polynomials have
enough common zeros. We examine a class of orthogonal
polynomials of two variables generated from that of one variable.
Received February 9, 1993 / Revised version received
January 18, 1994 相似文献
4.
Summary. This paper generalizes the idea of approximation on sparse grids to discrete differential forms that include )- and -conforming mixed finite element spaces as special cases. We elaborate on the construction of the spaces, introduce suitable
nodal interpolation operators on sparse grids and establish their approximation properties. We discuss how nodal interpolation
operators can be approximated. The stability of -conforming finite elements on sparse grids, when used to approximate second order elliptic problems in mixed formulation,
is investigated both theoretically and in numerical experiments.
Received November 2, 2000 / Revised version received October 23, 2001 / Published online January 30, 2002
This work was supported by DFG.
This paper is dedicated to Ch. Zenger on the occasion of his 60th birthday. 相似文献
5.
Summary. There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement
formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one
to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector
fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the
analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional
difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable.
We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for
the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions
to the construction of such finite element spaces, which account for the paucity of elements available.
Received January 10, 2001 / Published online November 15, 2001 相似文献
6.
Summary.
The interpolation theorem for convex quadrilateral
isoparametric finite elements is proved in the case when the condition
is not satisfied, where is the
diameter of the element and
is the radius of an
inscribed circle in .
The interpolation error is
in the -norm and
in the
-norm provided
that the interpolated function belongs to
. In the case when
the long sides of the quadrilateral
are parallel the constants
appearing in the estimates are evaluated.
Received
September 1993 / Revised version received March 6, 1995 相似文献
7.
Superconvergence and a posteriori error estimation for
triangular mixed finite elements 总被引:5,自引:0,他引:5
Jan H. Brandts 《Numerische Mathematik》1994,68(3):311-324
Summary. In this paper,we prove superconvergence results for the vector
variable when lowest order triangular mixed finite elements of
Raviart-Thomas type [17] on uniform triangulations are used,
i.e., that the -distance between the
approximate solution and a suitable projection of the real solution
is of higher order than the -error. We
prove
results for both Dirichlet and Neumann boundary conditions. Recently,
Duran [9] proved similar results for rectangular mixed finite
elements, and superconvergence along the Gauss-lines for rectangular
mixed finite elements was considered by Douglas, Ewing, Lazarov and
Wang in [11], [8] and [18]. The triangular case
however needs some extra effort. Using the superconvergence results,
a simple postprocessing of the approximate solution will give an
asymptotically exact a posteriori error estimator for the
-error in the approximation of the vector variable.
Received December 6, 1992 / Revised
version received October 2, 1993 相似文献
8.
Summary. We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the
exterior of a bounded closed domain in , . We describe a procedure to generate a sequence of bounded computational domains , , more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm
by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples
show the optimal order of convergence.
Received July 8, 1997 /Revised version received October 23, 1997 相似文献
9.
Summary. Macro-elements of smoothness on Clough-Tocher triangle splits are constructed for all . These new elements are improvements on elements constructed in [11] in that (disproving a conjecture made there) certain
unneeded degrees of freedom have been removed. Numerical experiments on Hermite interpolation with the new elements are included.
Received September 6, 2000 / Revised version received November 15, 2000 / Published online July 25, 2001 相似文献
10.
Summary. We compare the robustness of three different low-order mixed methods that have been proposed for plate-bending problems:
the so-called MITC, Arnold-Falk and Arnold-Brezzi elements. We show that for free plates, the asymptotic rate of convergence
in the presence of quasiuniform meshes approaches the optimal O(h) for MITC elements as the thickness approaches 0, but only approaches for the latter two. We accomplish this by establishing lower bounds for the error in the rotation. The deterioration occurs due to a consistency error associated with the boundary layer
– we show how a modification of the elements at the boundary can fix the problem. Finally, we show that the Arnold-Brezzi
element requires extra regularity for the convergence of the limiting (discrete Kirchhoff) case, and show that it fails to
converge in the presence of point loads.
Received June 9, 1998 / Published online December 6, 1999 相似文献
11.
Convergent adaptive finite elements for the nonlinear Laplacian 总被引:3,自引:3,他引:0
Andreas Veeser 《Numerische Mathematik》2002,92(4):743-770
Summary. The numerical solution of the homogeneous Dirichlet problem for the p-Laplacian, , is considered. We propose an adaptive algorithm with continuous piecewise affine finite elements and prove that the approximate
solutions converge to the exact one. While the algorithm is a rather straight-forward generalization of those for the linear
case p=2, the proof of its convergence is different. In particular, it does not rely on a strict error reduction.
Received December 29, 2000 / Revised version received August 30, 2001 / Published online December 18, 2001
RID="*"
ID="*" Current address: Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy; e-mail: veeser@mat.unimi.it 相似文献
12.
Summary. In this paper, we consider the problem of designing plate-bending elements which are free of shear locking. This phenomenon is known to afflict several elements for the Reissner-Mindlin plate model when the thickness of the plate is small, due to the inability of the approximating subspaces to satisfy the Kirchhoff constraint. To avoid locking, a “reduction operator” is often applied to the stress, to modify the variational formulation and reduce
the effect of this constraint. We investigate the conditions required on such reduction operators to ensure that the approximability
and consistency errors are of the right order. A set of sufficient conditions is presented, under which optimal errors can
be obtained – these are derived directly, without transforming the problem via a Hemholtz decomposition, or considering it
as a mixed method. Our analysis explicitly takes into account boundary layers and their resolution, and we prove, via an asymptotic
analysis, that convergence of the finite element approximations will occur uniformly as , even on quasiuniform meshes. The analysis is carried out in the case of a free boundary, where the boundary layer is known
to be strong. We also propose and analyze a simple post-processing scheme for the shear stress. Our general theory is used
to analyze the well-known MITC elements for the Reissner-Mindlin plate. As we show, the theory makes it possible to analyze
both straight and curved elements. We also analyze some other elements.
Received June 19, 1995 相似文献
13.
Summary. Using a method based on quadratic nodal spline interpolation, we define a quadrature rule with respect to arbitrary nodes,
and which in the case of uniformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which
is an important example of a trapezoidal rule with endpoint corrections. The resulting weights are explicitly calculated,
and Peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown
to grow at most linearly with respect to the mesh ratio parameter. Specializing these error estimates to the case of uniform
nodes, we deduce non-optimal order error constants for the Lacroix rule, which are significantly smaller than those calculated
by cruder methods in previous work, and which are shown here to compare favourably with the corresponding error constants
for the Simpson rule.
Received July 27, 1998/ Revised version received February 22, 1999 / Published online January 27, 2000 相似文献
14.
Nonlinear Galerkin methods and mixed finite elements:
two-grid algorithms for the Navier-Stokes equations 总被引:14,自引:0,他引:14
Summary.
A nonlinear Galerkin method using mixed finite
elements is presented for the two-dimensional
incompressible Navier-Stokes equations. The
scheme is based on two finite element spaces
and for the approximation of the velocity,
defined respectively on one coarse grid with grid
size and one fine grid with grid size and
one finite element space for the approximation
of the pressure. Nonlinearity and time
dependence are both treated on the coarse space.
We prove that the difference between the new
nonlinear Galerkin method and the standard
Galerkin solution is of the order of $H^2$, both in
velocity ( and pressure norm).
We also discuss a penalized version of our algorithm
which enjoys similar properties.
Received October 5, 1993 / Revised version received November
29, 1993 相似文献
15.
Motivated by earlier considerations of interval interpolation problems as well as a particular application to the reconstruction
of railway bridges, we deal with the problem of univariate convexity preserving interval interpolation. To allow convex interpolation,
the given data intervals have to be in (strictly) convex position. This property is checked by applying an abstract three-term
staircase algorithm, which is presented in this paper. Additionally, the algorithm provides strictly convex ordinates belonging
to the data intervals. Therefore, the known methods in convex Lagrange interpolation can be used to obtain interval interpolants.
In particular, we refer to methods based on polynomial splines defined on grids with additional knots.
Received September 22, 1997 / Revised version received May 26, 1998 相似文献
16.
Summary. A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic
obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits
optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is
explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results
to double obstacle problems are briefly discussed.
Received June 19, 1998 / Published online December 6, 1999 相似文献
17.
Summary. The use of mixed finite element methods is well-established in the numerical approximation of the problem of nearly incompressible elasticity, and its limit, Stokes flow. The question of stability over curved elements for such methods is of particular significance in the p version, where, since the element size remains fixed, exact representation of the curved boundary by (large) elements is often used. We identify a mixed element which we show to be optimally stable in both p and h refinement over curvilinear meshes. We prove optimal p version (up to ) and h version (p = 2, 3) convergence for our element, and illustrate its optimality through numerical experiments. Received August 25, 1998 / Revised version received February 16, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000 相似文献
18.
Summary. We consider the isoparametric transformation, which maps a given reference element onto a global element given by its vertices,
for multi-linear finite elements on pyramids and prisms. We present easily computable conditions on the position of the vertices,
which ensure that the isoparametric transformation is bijective.
Received May 7, 1999 / Revised version received April 28, 2000 / Published online December 19, 2000 相似文献
19.
Summary. In this paper, we derive the optimal error bounds for the stabilized MITC3 element [3], the MIN3 type element [7] and the T3BL element [8]. In this way we have
solved the problem proposed recently in [5] in a positive manner. Moreover, we estimate the difference between stabilized
MITC3 and MIN3 and show it is of order uniform in the plate thickness.
Received May 31, 2000 / Revised version received April 2, 2001 / Published online September 19, 2001 相似文献
20.
Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to
a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in
elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed
in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis
of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with
a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical
examples are included.
Received February 21, 1995 / Revised version received December 21, 1995 相似文献