Results 1  10
of
30
GUARANTEED AND ROBUST DISCONTINUOUS GALERKIN A POSTERIORI ERROR ESTIMATES FOR CONVECTION–DIFFUSION–REACTION PROBLEMS
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2008
"... We propose and study a posteriori error estimates for convection–diffusion–reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interiorpenalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to anal ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
We propose and study a posteriori error estimates for convection–diffusion–reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interiorpenalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using H(div, Ω)conforming diffusive and convective flux reconstructions, thereby extending previous work on pure diffusion problems. The resulting estimates are semirobust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Péclet and Damköhler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skewsymmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical
Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems
 COMPUT. METHODS APPL. MECH. ENGRG
, 2009
"... We derive a posteriori error estimates for a class of secondorder monotone quasilinear diffusiontype problems approximated by piecewise affine, continuous finite elements. Our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual, a ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
We derive a posteriori error estimates for a class of secondorder monotone quasilinear diffusiontype problems approximated by piecewise affine, continuous finite elements. Our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual, as well as a global error lower bound, up to a generic constant independent of the nonlinear operator. They are thus fully robust with respect to the nonlinearity, thanks to the choice of the error measure. They are also locally efficient, albeit in a different norm, and hence suitable for adaptive mesh refinement. Moreover, they allow to distinguish, estimate separately, and compare the discretization and linearization errors. Hence, the iterative (Newton–Raphson, quasiNewton) linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error. This can lead to important computational savings, as performing an excessive number of unnecessary linearization iterations can be avoided. Numerical experiments for the pLaplacian illustrate the theoretical developments.
A POSTERIORI ERROR ESTIMATION BASED ON POTENTIAL AND FLUX RECONSTRUCTION FOR THE HEAT EQUATION ∗
"... Abstract. We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler sch ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
Abstract. We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler scheme in time. Our estimates are based on a H 1conforming reconstruction of the potential, continuous and piecewise affine in time, and a locally conservative H(div)conforming reconstruction of the flux, piecewise constant in time. They yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative. Localintime lower bounds are also derived; for nonconforming methods on timevarying meshes, the lower bounds require a mild parabolictype constraint on the meshsize.
ROBUST APOSTERIORI ESTIMATOR FOR ADVECTIONDIFFUSIONREACTION PROBLEMS
"... Abstract. We propose an almostrobust residualbased aposteriori estimator for the advectiondiffusionreaction model problem. The theory is developed in the onedimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We propose an almostrobust residualbased aposteriori estimator for the advectiondiffusionreaction model problem. The theory is developed in the onedimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful aposteriori estimator, robust up to a √ log(Pe) factor, where Pe is the global Péclet number of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature. We also consider a possible twodimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the onedimensional setting. 1.
A POSTERIORI ENERGYNORM ERROR ESTIMATES FOR ADVECTIONDIFFUSION EQUATIONS APPROXIMATED BY WEIGHTED INTERIOR PENALTY METHODS
"... We propose and analyze a posteriori energynorm error estimates for weighted interior penalty discontinuous Galerkin approximations to advectiondiffusionreaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tens ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
We propose and analyze a posteriori energynorm error estimates for weighted interior penalty discontinuous Galerkin approximations to advectiondiffusionreaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a nonconforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfürth on continuous finite elements, that the constants are bounded by the square root of the local Péclet number.
A unified framework for a posteriori error estimation for the Stokes problem
 NUMERISCHE MATHEMATIK
, 2011
"... In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on [H10 (Ω)] dconforming velocity reconstruction and H(div, Ω)conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper boun ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on [H10 (Ω)] dconforming velocity reconstruction and H(div, Ω)conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix–Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. Numerical experiments illustrate the theoretical developments.
A posteriori error estimates for finite element exterior calculus: The de rham complex. arXiv:1203.0803v3
, 2012
"... Abstract. Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther [4] inclu ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther [4] includes a welldeveloped theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for ArnoldFalkWinther mixed finite element methods for Hodgede Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian. 1.
Residualbased a posteriori error estimates of nonconforming finite element method for elliptic problem with Dirac delta source terms
 Science in China Series A: Mathematics
"... ar ..."
(Show Context)
A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods
"... Galerkin methods ..."