共查询到20条相似文献,搜索用时 31 毫秒
1.
Rodolfo Araya Gabriel R. Barrenechea Abner Poza 《Journal of Computational and Applied Mathematics》2008
In this work we present an adaptive strategy (based on an a posteriori error estimator) for a stabilized finite element method for the Stokes problem, with and without a reaction term. The hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble-like functions. An equivalence result between the norm of the finite element error and the estimator is given, where the dependence of the constants on the physics of the problem is explicited. Several numerical results confirming both the theoretical results and the good performance of the estimator are given. 相似文献
2.
Barbara I. Wohlmuth 《Numerische Mathematik》1999,84(1):143-171
Summary. A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite
element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves
to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual
based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained
in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably
reduced if the special structure of the Lagrange multiplier is taken into account.
Received July 18, 1997 / Revised version received July 27, 1998 / Published online September 7, 1999 相似文献
3.
This paper presents an a posteriori error analysis for the linear finite element approximation of the Signorini problem in
two space dimensions. A posteriori estimations of residual type are defined and upper and lower bounds of the discretization
error are obtained. We perform several numerical experiments in order to compare the convergence of the terms in the error
estimator with the discretization error. 相似文献
4.
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
5.
Reliable and efficient residual-based a posteriori error estimates are established for the stabilised locking-free finite element methods for the Reissner-Mindlin plate model. The error is estimated by a computable error estimator from above and below up to multiplicative constants that do neither depend on the mesh-size nor on the plate's thickness and are uniform for a wide range of stabilisation parameter. The error is controlled in norms that are known to converge to zero in a quasi-optimal way. An adaptive algorithm is suggested and run for improving the convergence rates in three numerical examples for thicknesses 0.1, .001 and .001. 相似文献
6.
Kwang-Yeon Kim Hyung-Chun Lee 《Journal of Computational and Applied Mathematics》2010,235(1):186-2912
In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results. 相似文献
7.
This paper presents a robust a posteriori residual error estimator for diffusion-convection-reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in Rd, d=2 or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results. 相似文献
8.
R. Verfürth 《Numerische Mathematik》1991,60(1):235-249
Summary We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator. 相似文献
9.
Kwang Y. Kim 《Numerische Mathematik》2005,101(1):121-142
In this work we propose and analyze a mixed finite volume method for the p-Laplacian problem which is based on the lowest order Raviart–Thomas element for the vector variable and the P1 nonconforming element for the scalar variable. It is shown that this method can be reduced to a P1 nonconforming finite element method for the scalar variable only. One can then recover the vector approximation from the computed scalar approximation in a virtually cost-free manner. Optimal a priori error estimates are proved for both approximations by the quasi-norm techniques. We also derive an implicit error estimator of Bank–Weiser type which is based on the local Neumann problems.This work was supported by the Post-doctoral Fellowship Program of Korea Science & Engineering Foundation (KOSEF). 相似文献
10.
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
11.
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
12.
Stefano Berrone 《Numerische Mathematik》2002,91(3):389-422
Summary. We derive a residual-based a posteriori error estimator for a stabilized finite element discretization of certain incompressible Oseen-like equations. We focus our
attention on the behaviour of the effectivity index and we carry on a numerical study of its sensitiveness to the problem
and mesh parameters. We also consider a scalar reaction-convection-diffusion problem and a divergence-free projection problem
in order to investigate the effects on the robustness of our a posteriori error estimator of the reaction-convection-diffusion phenomena and, separately, of the incompressibility constraint.
Received March 21, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001 相似文献
13.
Willy Dörfler 《Numerische Mathematik》1996,73(4):419-448
Summary.
We prove an a posteriori error estimate for the linear
time-dependent Schr?dinger equation in .
From this, we derive a residual
based local error estimator that allows us to adjust the mesh and the
time step size in order to obtain a numerical solution with a prescribed
accuracy. As a special feature, the error estimator controls
localization and size of the finite computational domain in each time
step. An algorithm is described to compute this solution and numerical
results in one space dimension are included.
Received March 17, 1995 相似文献
14.
Mark Ainsworth 《Journal of Computational and Applied Mathematics》2010,234(9):2618-2632
We give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming, non-conforming, mixed and discontinuous finite element schemes. Whilst the details of the actual estimator are rather different for each particular scheme, there is nonetheless a common underlying structure at work in all cases. We aim to illustrate this structure by treating conforming, non-conforming and discontinuous finite element schemes in a single framework. In taking a rather general viewpoint, some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but, in return, we hope to allow the reader to ‘see the wood despite the trees’. 相似文献
15.
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
16.
JaEun Ku 《Journal of Computational and Applied Mathematics》2010,235(1):293-836
In this paper, we propose a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method. In particular, we propose an a posteriori error estimator for when one is interested in where . Our a posteriori error estimators are obtained by assigning proper weight (in terms of local mesh size hT) to the terms of the least-squares functional. An a posteriori error analysis yields reliable and efficient estimates based on residuals. Numerical examples are presented to show the effectivity of our error estimators. 相似文献
17.
In this paper, we present a posteriori error analysis for the nonconforming Morley element of the fourth order elliptic equation.
We propose a new residual-based a posteriori error estimator and prove its reliability and efficiency. These results refine
those of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) by dropping two edge jump terms in both the energy norm of
the error and the estimator, and those of Wang and Zhang (Local a priori and a posteriori error estimates of finite elements
for biharmonic equation, Research Report, 13, 2006) by showing the efficiency in the sense of Verfürth (A review of a posteriori
error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, New York, 1996). Moreover, the normal component in
the estimators of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) and Wang and Zhang (Local a priori and a posteriori
error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) is dropped, and therefore only the
tangential component of the stress on each edge comes into the estimator. In addition, we generalize these results to three
dimensional case. 相似文献
18.
G. Lube 《Numerische Mathematik》1992,61(1):335-357
Summary We extend the analysis of the streamline diffusion finite element method to quasilinear elliptic problems of second order. An existence theorem and error estimates are given in the case of branches of nonsingular solutions following a recent abstract approach in [12, 13, 26]. 相似文献
19.
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
20.
Summary This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)]. 相似文献