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1.
Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner–Mindlin
plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by
two parts: one measures the local approximability of the exact solution by the finite element space and the other the global
approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold–Falk
element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal
order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
2.
In this paper, we focus on a local superconvergence analysis of the finite element method for the Stokes equations by local projections. The local and global superconvergence results of finite element solutions are provided for the Stokes problem under some corresponding regularity assumptions. Conclusion can be drawn that the local superconvergence has advantages over the global superconvergence in two important aspects. On the one hand, it offsets theoretical limitation in practical applications. On the other hand, interior estimates are derived on the base of local properties of the domain without global smoothness for the exact solution and prior regularity of the problem globally over the whole domain. 相似文献
3.
Bhupen Deka 《Journal of Computational and Applied Mathematics》2010,234(2):605-612
The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings. 相似文献
4.
This paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible
Navier–Stokes equations. Incompressibility and convective effects are both stabilized adding an interior penalty term giving
L
2-control of the jump of the gradient of the approximate solution over the internal faces. Using continuous equal-order finite
elements for both velocities and pressures, in a space semi-discretized formulation, we prove convergence of the approximate
solution. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations,
provided the exact solution is smooth. 相似文献
5.
Takuya Tsuchiya 《Numerische Mathematik》1999,84(1):121-141
Summary. Finite element solutions of strongly nonlinear elliptic boundary value problems are considered. In this paper, using the
Kantorovich theorem, we show that, if the Fréchet derivative of the nonlinear operator defined by the boundary value problem
is an isomorphism at an exact solution, then there exists a locally unique finite element solution near the exact solution.
Moreover, several a priori error estimates are obtained.
Received March 2, 1998 / Published online September 7, 1999 相似文献
6.
In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions.
In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry
and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element
method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases
when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments
confirm the theoretical results and show the potential of our proposed method. 相似文献
7.
Erkki Heikkola Sanna Mönkölä Anssi Pennanen Tuomo Rossi 《Journal of Computational and Applied Mathematics》2007
We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced. 相似文献
8.
W. Spann 《Numerische Mathematik》1994,69(1):103-116
Summary.
An abstract error estimate for the approximation of semicoercive variational
inequalities is obtained provided a certain condition holds for the exact
solution. This condition turns out to be necessary as is demonstrated
analytically and numerically. The results are applied to the finite element
approximation of Poisson's equation with Signorini boundary conditions
and to the obstacle problem for the beam with no fixed boundary conditions.
For second order variational inequalities the condition is always satisfied,
whereas for the beam problem the condition holds if the center of forces
belongs to the interior of the convex hull of the contact set. Applying the error
estimate yields optimal order of convergence in terms of the mesh size
.
The numerical convergence rates observed are in good agreement with the
predicted ones.
Received August 16, 1993 /
Revised version received March 21, 1994 相似文献
9.
Numerical verification of solutions for variational inequalities 总被引:1,自引:0,他引:1
In this paper, we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities.
This technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations.
Using the finite element approximations and explicit a priori error estimates for obstacle problems, we present an effective
verification procedure that through numerical computation generates a set which includes the exact solution. Further, a numerical
example for an obstacle problem is presented.
Received October 28,1996 / Revised version received December 29,1997 相似文献
10.
Silvia Bertoluzza 《Numerische Mathematik》1997,78(1):1-20
Summary. In this paper we derive an interior estimate for the Galerkin method with wavelet-type basis. Such an estimate follows from
interior Galerkin equations which are common to a class of methods used in the solution of elliptic boundary value problems.
We show that the error in an interior domain can be estimated with the best order of accuracy possible, provided the solution is sufficiently regular in a slightly larger domain, and that an estimate of the same order exists for the error in a weaker
norm (measuring the effects from outside the domain ). Examples of the application of such an estimate are given for different problems.
Received May 17, 1995 / Revised version received April 26, 1996 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.
Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.The work was partially supported by ONR Contract N00014-77-C-0623 相似文献
13.
Xiao-bo Liu 《计算数学(英文版)》1999,17(5):475-494
1.IntroductionInteriorerrorestimatesforfiniteelementdiscretizations(conforming)werefirstintroducedbyNitscheandSchatz[14]forsecondorderscalarellipticequationsin1974.Theyprovedthatthelocalaccuracyofthefiniteelementapproximationisboundedintermsoftwofact... 相似文献
14.
Superconvergence for rectangular mixed finite elements 总被引:4,自引:0,他引:4
Ricardo Durán 《Numerische Mathematik》1990,58(1):287-298
Summary In this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL
2 between the approximate solution and a projection of the exact one is of higher order than the error itself.This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing. 相似文献
15.
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
16.
Jinn-Liang Liu Werner C. Rheinboldt 《Numerical Functional Analysis & Optimization》2013,34(3-4):335-356
A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach. 相似文献
17.
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
18.
In this paper, a discontinuous Galerkin method for the two-dimensional time-harmonic Maxwell equations in composite materials is presented. The divergence constraint is taken into account by a regularized variational formulation and the tangential and normal jumps of the discrete solution at the element interfaces are penalized. Due to an appropriate mesh refinement near exterior and interior corners, the singular behaviour of the electromagnetic field is taken into account. Optimal error estimates in a discrete energy norm and in the L2-norm are proved in the case where the exact solution is singular. 相似文献
19.
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
20.
J. Steinbach 《Numerical Functional Analysis & Optimization》2013,34(9-10):1041-1066
General finite volume approximations in two and three space dimensions are studied for the discretization of interior and boundary obstacle problems with mixed boundary conditions. First order convergence of the finite volume (box) solution is shown in the energy norm. Based on a discrete maximum principle there are proposed two penalization techniques for the solution of the finite volume inequalities. By coupling of discretization and penalty parameters the overall error is analyzed. The iterative solution of the penalty problems is discussed. Finally, results of numerical experiments are presented to illustrate the convergence behaviour between the exact, the box and the penalty solutions. 相似文献