Results 1  10
of
47
Residual Based A Posteriori Error Estimators For Eddy Current Computation
, 1999
"... We consider H(curl;\Omega\Gamma3932/608 problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a ..."
Abstract

Cited by 51 (7 self)
 Add to MetaCart
(Show Context)
We consider H(curl;\Omega\Gamma3932/608 problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtztype decomposition of the error into an irrotational part and a weakly solenoidal part.
Robust a posteriori error estimation for nonconforming finite element approximation
 SIAM J. Numer. Anal
"... Abstract. We obtain a computable a posteriori error bound on the broken energy norm of the error in the Fortin–Soulie finite element approximation of a linear second order elliptic problem with variable permeability. This bound is shown to be efficient in the sense that it also provides a lower boun ..."
Abstract

Cited by 36 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We obtain a computable a posteriori error bound on the broken energy norm of the error in the Fortin–Soulie finite element approximation of a linear second order elliptic problem with variable permeability. This bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms. The estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error. 1.
A POSTERIORI ERROR ESTIMATES FOR MAXWELL EQUATIONS
"... Maxwell equations are posed as variational boundary value problems in the function space H(curl) and are discretized by Nédélec finite elements. In [4], a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove th ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
Maxwell equations are posed as variational boundary value problems in the function space H(curl) and are discretized by Nédélec finite elements. In [4], a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasiinterpolation operators introduced recently in [22]. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.
Error reduction and convergence for an adaptive mixed finite element method
 Mathematics of Computation
, 2005
"... Abstract. An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction prope ..."
Abstract

Cited by 31 (9 self)
 Add to MetaCart
(Show Context)
Abstract. An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart–Thomas finite element method with a reduction factor ρ<1 uniformly for the L 2 norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasiorthogonality estimate. The proof does not rely on duality or on regularity. 1.
A POSTERIORI ERROR ESTIMATES FOR LOWESTORDER MIXED FINITE ELEMENT DISCRETIZATIONS OF CONVECTIONDIFFUSIONREACTION EQUATIONS
, 2007
"... We establish residual a posteriori error estimates for lowestorder Raviart–Thomas mixed finite element discretizations of convectiondiffusionreaction equations on simplicial meshes in two or three space dimensions. The upwindmixed scheme is considered as well, and the emphasis is put on the pres ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
We establish residual a posteriori error estimates for lowestorder Raviart–Thomas mixed finite element discretizations of convectiondiffusionreaction equations on simplicial meshes in two or three space dimensions. The upwindmixed scheme is considered as well, and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusiondispersion tensor and on a possible convection dominance. Global upper bounds for the approximation error in the energy norm are derived, where in particular all constants are evaluated explicitly, so that the estimators are fully computable. Our estimators give local lower bounds for the error as well, and they hold from the cases where convection or reaction are not present to convection or reactiondominated problems; we prove that their local efficiency depends only on local variations in the coefficients and on the local Péclet number. Moreover, the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies. The main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element, known in the mixed finite element method, and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem. Numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators.
Convergence and optimality of adaptive mixed finite element methods
, 2009
"... The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
(Show Context)
The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.
A Posteriori Error Control In LowOrder Finite Element Discretisations Of Incompressible Stationary Flow Problems
 Math. Comput
, 1999
"... . Computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and nonconforming finite element methods. A refined residualbased error estimate generalises th ..."
Abstract

Cited by 20 (7 self)
 Add to MetaCart
. Computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and nonconforming finite element methods. A refined residualbased error estimate generalises the works of Verfurth, Dari, Duran & Padra, Bao & Barrett. As a consequence, reliable and efficient averaging estimates can be established on unstructured grids. The symmetric formulation of the incompressible flow problem models certain nonNewtonian flow problems and the Stokes problem with mixed boundary conditions. A Helmholtz decomposition avoids any regularity or saturation assumption in the mathematical error analysis. Numerical experiments for the partly nonconforming method analysed by Kouhia and Stenberg indicate efficiency of related adaptive meshrefining algorithms. 1. Introduction Adaptive finite element methods play an important practical role in computational fluid dynamics. T...
A posteriori error estimates for the mortar mixed finite element method
 SIAM J. Numer. Anal
"... Abstract. Several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived. A residualbased estimator provides optimal upper and lower bounds for the pressure error. An efficient and reliable estimator for the velocity and mortar pressure error ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
(Show Context)
Abstract. Several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived. A residualbased estimator provides optimal upper and lower bounds for the pressure error. An efficient and reliable estimator for the velocity and mortar pressure error is also derived, which is based on solving local (element) problems in a higherorder space. The interface fluxjump term that appears in the estimators can be used as an indicator for driving an adaptive process for the mortar grids only.
A POSTERIORI ERROR ANALYSIS FOR LOCALLY CONSERVATIVE MIXED METHODS
"... Abstract. In this work we present a theoretical analysis for a residualtype error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart–Thomas mixed finite element method working in meshdependent norms. We improve and extend th ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
(Show Context)
Abstract. In this work we present a theoretical analysis for a residualtype error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart–Thomas mixed finite element method working in meshdependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoiding the saturation assumption made by Braess and Verfürth. Our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries. The main results are applied to the P 1 nonconforming finite element method and the interior penalty discontinuous Galerkin method as well as the mixed finite element method. 1.
Energy norm a posteriori error estimates for mixed finite element methods
, 2004
"... Abstract. This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different w ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
(Show Context)
Abstract. This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different ways: under a saturation assumption and using a Helmholtz decomposition for vector fields. 1.