共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
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
4.
The global error of numerical approximations for symmetric positive systems in the sense of Friedrichs is decomposed into
a locally created part and a propagating component. Residual-based two-sided local a posteriori error bounds are derived for
the locally created part of the global error. These suggest taking the -norm as well as weaker, dual norms of the computable residual as local error indicators. The dual graph norm of the residual
is further bounded from above and below in terms of the norm of where h is the local mesh size. The theoretical results are illustrated by a series of numerical experiments.
Received January 10, 1997 / Revised version received March 5, 1998 相似文献
5.
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
6.
R. Verfürth 《Numerische Mathematik》1998,80(4):641-663
Summary. We derive a posteriori error estimators for convection-diffusion equations with dominant convection. 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
only depends on the local mesh-Peclet number. The estimators are either based on the evaluation of local residuals or on the
solution of discrete local Dirichlet or Neumann problems.
Received February 10, 1997 / Revised version received November 4, 1997 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.
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
9.
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
10.
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
11.
Summary.
An adaptive finite element method for the calculation of transonic potential
flows was developed. An error indicator based on first order finite differences
of gradients is introduced as a local error estimator. It measures second order
distributional derivatives. Estimates involving
this error estimator, a residual and the error are given. The error estimator
can be used as a criterion for mesh refinement. We also give some computational
results.
Received September 16, 1993 / Revised version received June
7, 1994 相似文献
12.
New anisotropic a priori error estimates 总被引:5,自引:0,他引:5
Summary. We prove a priori anisotropic estimates for the and interpolation error on linear finite elements. The full information about the mapping from a reference element is employed
to separate the contribution to the elemental error coming from different directions. This new error estimate does not require the “maximal angle condition”. The analysis has been carried out for the 2D case, but may
be extended to three dimensions. Numerical experiments have been carried out to test our theoretical results.
Received March 3, 2000 / Revised version received June 27, 2000 / Published online April 5, 2001 相似文献
13.
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
14.
A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations 总被引:1,自引:0,他引:1
Mario Ohlberger 《Numerische Mathematik》2001,87(4):737-761
Summary. This paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived
in the -norm in the situation, where the diffusion parameter is smaller or comparable to the mesh size. Numerical experiments underline the theoretical results.
Received February 25, 1999 / Revised version received July 6, 1999 / Published online August 2, 2000 相似文献
15.
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
16.
Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations
on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate
for the numerical solution is obtained.
Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000 相似文献
17.
K. Segeth 《Numerische Mathematik》1999,83(3):455-475
Summary. Convergence of a posteriori error estimates to the true error for the semidiscrete finite element method of lines is shown
for a nonlinear parabolic initial-boundary value problem.
Received June 15, 1997 / Revised version received May 15, 1998 / Published online: June 29, 1999 相似文献
18.
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
19.
Summary. We develop the general a priori error analysis of residual-free bubble finite element approximations to non-self-adjoint
elliptic problems of the form subject to homogeneous Dirichlet boundary condition, where A is a symmetric second-order elliptic operator, C is a skew-symmetric first-order differential operator, and is a positive parameter. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements
of degree .
Received October 1, 1998/ Revised version received April 6, 1999 / Published online January 27, 2000 相似文献
20.
Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations 总被引:1,自引:0,他引:1
Summary. The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows
the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection
of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference
between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation
is shown to be stable and optimally convergent, both in a norm associated to the problem and in the norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear
case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in
spite of the fact that the present method is only a minor modification of this.
Received May 25, 1998 / Revised version received August 31, 1999 / Published online July 12, 2000 相似文献