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17
Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids
 Part II: Higher order FEM, Math. Comp., posted on February 4, 2002, PII S
"... Abstract. Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conformi ..."
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Cited by 35 (8 self)
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Abstract. Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given righthand sides. 1.
Convergence of adaptive finite element methods in computational mechanics
 Proceedings of the Sixth World Congress on Computational Mechanics
, 2004
"... Abstract. The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solu ..."
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Cited by 7 (3 self)
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Abstract. The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R−linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropickinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule. 1.
All firstorder averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable
"... Abstract. All firstorder averaging or gradientrecovery operators for lowestorder finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain Ω in Rd. Given a piecewise constant discrete flux ph ∈ Ph ..."
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Cited by 6 (1 self)
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Abstract. All firstorder averaging or gradientrecovery operators for lowestorder finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain Ω in Rd. Given a piecewise constant discrete flux ph ∈ Ph (that is the gradient of a discrete displacement) as an approximation to the unknown exact flux p (that is the gradient of the exact displacement), recent results verify efficiency and reliability of ηM: = min{‖ph − qh‖L2 (Ω) : qh ∈Qh} in the sense that ηM is a lower and upper bound of the flux error ‖p−ph ‖ L2 (Ω) up to multiplicative constants and higherorder terms. The averaging space Qh consists of piecewise polynomial and globally continuous finite element functions in d components with carefully designed boundary conditions. The minimal value ηM is frequently replaced by some averaging operator A: Ph → Qh applied within a simple postprocessing to ph. Theresultqh: = Aph ∈Qh provides a reliable error bound with ηM ≤ ηA: = ‖ph − Aph‖L2 (Ω). This paper establishes ηA ≤ Ceff ηM and so equivalence of ηM and ηA. This implies efficiency of ηA for a large class of patchwise averaging techniques which includes the ZZgradientrecovery technique. The bound Ceff ≤ 3.88 established for tetrahedral P1 finite elements appears striking in that the shape of the elements does not enter: The equivalence ηA ≈ ηM is robust with respect to anisotropic meshes. The main arguments in the proof are Ascoli’s lemma, a strengthened Cauchy inequality, and elementary calculations with mass matrices. 1.
A Posteriori Error Estimates For Finite Volume Element Approximations Of ConvectionDiffusionReaction Equations
 Comput. Geosci
, 2002
"... We present the results of a study on a posteriori error control strategies for finite volume element approximations of second order elliptic differential equations. ..."
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Cited by 6 (1 self)
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We present the results of a study on a posteriori error control strategies for finite volume element approximations of second order elliptic differential equations.
ResidualBased A Posteriori Error Estimate For A Mixed ReissnerMindlin Plate Finite Element Method
, 2000
"... Reliable and efficient residualbased a posteriori error estimates are established for the stabilised lockingfree finite element methods for the ReissnerMindlin plate model. The error is estimated by a computable error estimator from above and below up to multiplicative constants that do neither d ..."
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Cited by 6 (3 self)
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Reliable and efficient residualbased a posteriori error estimates are established for the stabilised lockingfree finite element methods for the ReissnerMindlin 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 meshsize 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 quasioptimal way.
HP–interpolation of non–smooth functions
 Newton Institute Preprint NI03050CPD
, 2003
"... Abstract. The quasiinterpolation operators of Clément and ScottZhang type are generalized to the hpcontext. New polynomial lifting and inverse estimates are presented as well. Key words. Clément interpolant, quasiinterpolation, hpFEM, polynomial inverse estimates AMS subject classifications. 65 ..."
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Cited by 5 (1 self)
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Abstract. The quasiinterpolation operators of Clément and ScottZhang type are generalized to the hpcontext. New polynomial lifting and inverse estimates are presented as well. Key words. Clément interpolant, quasiinterpolation, hpFEM, polynomial inverse estimates AMS subject classifications. 65N30, 65N35, 65N50 1. Introduction. Quasiinterpolation operators
Convergence of a simple adaptive finite element method for . . .
, 2007
"... We prove convergence and optimal complexity of an adaptive finite element algorithm for a model problem of optimal control. Following previous work, our algorithm is based on an adaptive marking strategy which compares a simple edge estimator with an oscillation term in each step of the algorithm i ..."
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Cited by 2 (1 self)
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We prove convergence and optimal complexity of an adaptive finite element algorithm for a model problem of optimal control. Following previous work, our algorithm is based on an adaptive marking strategy which compares a simple edge estimator with an oscillation term in each step of the algorithm in order to adapt the marking of cells.
CONVERGENCE OF ADAPTIVE FEM FOR A CLASS OF DEGENERATE CONVEX MINIMIZATION PROBLEMS
"... Abstract. A class of degenerate convex minimization problems allows for some adaptive finite element method (AFEM) to compute strongly converging stress approximations. The algorithm AFEM consists of successive loops of the form SOLVE → ESTIMATE → MARK → REFINE and employs the bulk criterion. The co ..."
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Cited by 1 (0 self)
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Abstract. A class of degenerate convex minimization problems allows for some adaptive finite element method (AFEM) to compute strongly converging stress approximations. The algorithm AFEM consists of successive loops of the form SOLVE → ESTIMATE → MARK → REFINE and employs the bulk criterion. The convergence in Lp ′ (Ω; Rm×n) relies on new sharp strict convexity estimates of degenerate convex minimization problems with