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1.
We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates. 相似文献
2.
Emmanuel Creusé 《Journal of Computational and Applied Mathematics》2010,234(10):2903-2915
We consider some (anisotropic and piecewise constant) diffusion problems in domains of R2, approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. 相似文献
3.
The classical a posteriori error estimates are mostly oriented to the use in the finite element h-methods while the contemporary higher-order hp-methods usually require new approaches in a posteriori error estimation. These methods hold a very important position among adaptive numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications. 相似文献
4.
The aim of this paper is to introduce residual type a posteriori error estimators for a Poisson problem with a Dirac delta source term, in L
p
norm and W1,p
seminorm. The estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence. 相似文献
5.
We derive residual based a posteriori error estimates of the flux in L
2-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such
as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least
squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the
displacement which gives optimal order. 相似文献
6.
Serge Nicaise Juliette Venel 《Journal of Computational and Applied Mathematics》2011,235(14):4272-4282
We perform the a posteriori error analysis of residual type of transmission problem with sign changing coefficients. According to Bonnet-BenDhia et al. (2010) [9], if the contrast is large enough, the continuous problem can be transformed into a coercive one. We further show that a similar property holds for the discrete problem for any regular meshes, extending the framework from Bonnet-BenDhia et al. [9]. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests. 相似文献
7.
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
8.
Ricardo H. Nochetto Alfred Schmidt Kunibert G. Siebert Andreas Veeser 《Numerische Mathematik》2006,104(4):515-538
We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature. 相似文献
9.
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
10.
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
11.
Numerical approximation of the coupled system of compressible miscible displacement problem in porous media is considered in this paper. A continuous in time discontinuous Galerkin scheme is developed. The symmetric interior penalty discontinuous Galerkin method is used to solve both the flow and transport equations. Upwind technique is used to treat the convection term in the transport equation. The hp-a priori error bounds are derived. 相似文献
12.
A posteriori error estimators for the Stokes equations 总被引:5,自引:0,他引:5
Summary We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.This work was accomplished at the Universität Heidelberg with the support of the Deutsche Forschungsgemeinschaft 相似文献
13.
A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617–637, 2005) and Carstensen
and Hu (J Numer Math 107(3):473–502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied
to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based
computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming
finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature.
The estimate recovers known results as well as provides new bounds to a number of schemes.
C. Carstensen and M. Jensen supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and
the Hausdorff Institute of Mathematics in Bonn, Germany.
C. Carstensen, T. Gudi, and M. Jensen supported by DST-DAAD (PPP-05) project no. 32307481. 相似文献
14.
E. Gekeler 《Numerische Mathematik》1978,30(4):369-383
Summary Backward differentiation methods up to orderk=5 are applied to solve linear ordinary and partial (parabolic) differential equations where in the second case the space variables are discretized by Galerkin procedures. Using a mean square norm over all considered time levels a-priori error estimates are derived. The emphasis of the results lies on the fact that the obtained error bounds do not depend on a Lipschitz constant and the dimension of the basic system of ordinary differential equations even though this system is allowed to have time-varying coefficients. It is therefore possible to use the bounds to estimate the error of systems with arbitrary varying dimension as they arise in the finite element regression of parabolic problems. 相似文献
15.
A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the Hellinger-Reissner principle. This quasi-optimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Explicit residual-based a posteriori error estimates for DMH are introduced and are mathematically shown to be locking-free, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities. 相似文献
16.
Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods 总被引:1,自引:0,他引:1
Martin Vohralík 《Numerische Mathematik》2008,111(1):121-158
We derive in this paper a posteriori error estimates for discretizations of convection–diffusion–reaction equations in two
or three space dimensions. Our estimates are valid for any cell-centered finite volume scheme, and, in a larger sense, for
any locally conservative method such as the mimetic finite difference, covolume, and other. We consider meshes consisting
of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces. We
allow for the cases of inhomogeneous and anisotropic diffusion–dispersion tensors and of convection dominance. The estimates
are established in the energy (semi)norm for a locally postprocessed approximate solution preserving the conservative fluxes
and are of residual type. They are fully computable (all occurring constants are evaluated explicitly) and locally efficient
(give a local lower bound on the error times an efficiency constant), so that they can serve both as indicators for adaptive
refinement and for the actual control of the error. They are semi-robust in the sense that the local efficiency constant only
depends on local variations in the coefficients and becomes optimal as the local Péclet number gets sufficiently small. Numerical
experiments confirm their accuracy.
This work was supported by the GdR MoMaS project “Numerical Simulations and Mathematical Modeling of Underground Nuclear Waste
Disposal”, PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France.
The main part of this work was carried out during the author’s post-doc stay at Laboratoire de Mathématiques, Analyse Numérique
et EDP, Université de Paris-Sud and CNRS, Orsay, France. 相似文献
17.
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
18.
In this paper, we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by Stokes equations. We obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods, which are the patch recovery technique and the least-squares surface fitting method. Our results are based on some regularity assumption for the Stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions. 相似文献
19.
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
20.
Giancarlo Sangalli 《Numerische Mathematik》2001,89(2):379-399
Summary. We develop the a posteriori error analysis for the RFB method, applied to the linear advection-diffusion problem: the numerical
error, measured in suitable norms, is estimated in terms of the numerical residual. The robustness is investiged, in the sense
that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error.
Received January 21, 2000 / Published online March 20, 2001 相似文献