共查询到20条相似文献,搜索用时 0 毫秒
1.
Superconvergence analysis and error expansion for the Wilson nonconforming finite element 总被引:8,自引:0,他引:8
Summary.
In this paper the Wilson nonconforming finite element is considered for
solving a class of two-dimensional second-order elliptic boundary value
problems. Superconvergence estimates and error expansions are obtained
for both uniform and non-uniform rectangular meshes. A new lower bound
of the error shows that the usual error estimates are optimal. Finally
a discussion on the error behaviour in negative norms shows that there
is generally no improvement in the order by going to weaker norms.
Received July 5, 1993 相似文献
2.
Summary. Both for the - and -norms, we prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on
the error of linear finite element methods on anisotropic triangular or tetrahedral meshes. We also show that, with a correct
scaling, edge residuals yield a robust error estimator for a singularly perturbed reaction-diffusion equation.
Received April 19, 1999 / Published online April 20, 2000 相似文献
3.
Mark Ainsworth 《Numerische Mathematik》1996,73(4):399-418
Summary.
The element residual method for a posteriori error estimation is analyzed
for degree finite element approximation on quadrilateral elements.
The influence of the choice of subspace used to solve the element residual
problem is studied. It is shown that the resulting estimators will be
consistent (or asymptotically exact) for all
if and only if
the mesh is parallel. Moreover, even if the mesh consists of rectangles, then
the estimators can be inconsistent when .
The results provide concrete guidelines for the selection of a posteriori
error estimators and establish the limits of their performance. In particular,
the use of the element residual method for high orders of approximation
(such as those arising in the - version finite element method) is
vindicated.
The mechanism behind the rather poor performance of the estimators is traced
back to the basic formulation of the residual problem. The investigations
reveal a deficiency in the formulation, leading, as it does, to spurious
modes in the true solution of the residual problem. The recommended choice
of subspaces may be viewed as being sufficient to guarantee that the spurious
modes are filtered out from the approximate solution while at the same time
retaining a sufficient degree of approximation to represent the true modes.
Received February 27, 1995 / Revised version
received June 7, 1995 相似文献
4.
Summary. A semidiscrete mixed finite element approximation to parabolic initial-boundary value problems is introduced and analyzed.
Superconvergence estimates for both pressure and velocity are obtained. The estimates for the errors in pressure and velocity
depend on the smoothness of the initial data including the limiting cases of data in and data in , for sufficiently large. Because of the smoothing properties of the parabolic operator, these estimates for large time levels
essentially coincide with the estimates obtained earlier for smooth solutions. However, for small time intervals we obtain
the correct convergence orders for nonsmooth data.
Received July 30, 1995 / Revised version received October 14, 1996 相似文献
5.
Gerd Kunert 《Numerische Mathematik》2000,86(3):471-490
Summary. A new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes. All considerations carry over to anisotropic triangular meshes with minor changes only.
The lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence,
a so-called matching function is defined, and its discussion shows it to be a useful tool. With its help anisotropic interpolation estimates and subsequently
the upper error bound are proven. Additionally it is pointed out how to treat Robin boundary conditions in a posteriori error
analysis on isotropic and anisotropic meshes. A numerical example supports the anisotropic error analysis.
Received April 6, 1999 / Revised version received July 2, 1999 / Published online June 8, 2000 相似文献
6.
Kunibert G. Siebert 《Numerische Mathematik》1996,73(3):373-398
Summary.
Besides an algorithm for local refinement, an a posteriori error
estimator is the basic tool of every adaptive finite element
method. Using information generated by such an error estimator the
refinement of the grid is controlled. For 2nd order elliptic
problems we present an error estimator for anisotropically refined
grids, like -d cuboidal and 3-d prismatic grids, that gives
correct information about the size of the error; additionally it
generates information about the direction into which some element
has to be refined to reduce the error in a proper way. Numerical
examples are presented for 2-d rectangular and 3-d prismatic grids.
Received March 15, 1994 / Revised version received June 3, 1994 相似文献
7.
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
8.
A posteriori error estimates for mixed FEM in elasticity 总被引:2,自引:0,他引:2
A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value
problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions
of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from
inverse estimates. The constants in both estimates are independent of the Lamé constant , and so locking phenomena for are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement.
Received July 17, 1997 相似文献
9.
Mark Ainsworth 《Numerische Mathematik》1998,80(3):325-362
Summary. A posteriori error estimators for fully discrete hierarchic modelling on thin domains are derived and are shown to provide
computable upper bounds on the discretization error and on the total error. The estimators are shown to be robust and do not
degenerate as the thickness of the domain tends to zero. If the discretization part of the error is negligible, the estimator
for the modelling error reduces to the one recently obtained for semi-discrete hierarchical modelling by Babuska and Schwab.
Received July 25, 1996 / Revised version received July 31, 1997 相似文献
10.
Summary.
We consider the finite element approximation of a
non-Newtonian flow, where the viscosity obeys a general law including
the Carreau or power law. For sufficiently regular solutions we prove
energy type error bounds for the velocity and pressure. These bounds
improve on existing results in the literature. A key step in the
analysis is to prove abstract error bounds initially in a quasi-norm,
which naturally arises in degenerate problems of this type.
Received May 25, 1993 / Revised version received January 11, 1994 相似文献
11.
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
12.
Summary. Computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model Poisson-problem
in two and three space dimensions. Besides a refined residual-based a posteriori error estimate, an averaging estimator is
established and an -estimate is included. The a posteriori error estimates are reliable and efficient; the proof of reliability relies on a Helmholtz
decomposition.
Received March 4, 1997 / Revised version received September 4, 2001 / Published online December 18, 2001 相似文献
13.
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
14.
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
15.
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
16.
Summary. Stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods. Stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties. It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed methods, that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem. To do so, we rely on the fact that some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present two classes of stabilisations for the inf-sup condition in mixed problems. We present many examples, most arising from the discretisation of flow problems. Section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems. Received August 9, 1999 / Revised version received May 19, 2000 / Published online March 20, 2001 相似文献
17.
In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their
discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete
flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting
the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas
vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence
free vector fields as s of the –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite
element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient
spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof.
Received November 4, 1996 / Revised version received February 2, 1998 相似文献
18.
Summary. Two new augmented Lagrangian formulations for mixed finite element schemes are presented. The methods lead, in some cases,
to an improvement in the order of the approximation. An error analysis is provided, together with some interesting examples
of applications.
Received July 27, 1994 / Revised version received November 17, 1995 相似文献
19.
Mikko Lyly 《Numerische Mathematik》2000,85(1):77-107
Summary. We consider three triangular plate bending elements for the Reissner-Mindlin model. The elements are the MIN3 element of
Tessler and Hughes [19], the stabilized MITC3 element of Brezzi, Fortin and Stenberg [5] and the T3BL element of Xu, Auricchio
and Taylor [2, 17, 20]. We show that the bilinear forms of the stabilized MITC3 and MIN3 elements are equivalent and that
their implementation may be simplified by using numerical integration of reduced order. The T3BL element is shown to be essentially
the same as the MIN3 and stabilized MITC3 elements with reduced integration. We finally introduce a general stabilized finite
element formulation which covers all three methods. For this class of methods we prove the stability and optimal convergence
properties.
Received November 4, 1996 / Revised version received May 29, 1997 / Published online January 27, 2000 相似文献
20.
R. Verfürth 《Numerische Mathematik》1998,78(3):479-493
We derive robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Here, robust means
that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio
of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor
on the perturbation parameter. The estimators are based either on the evaluation of local residuals or on the solution of
discrete local Dirichlet or Neumann problems.
Received June 5, 1996 相似文献