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Analysis of Recovery Type A Posteriori Error Estimators for Mildly Structured Grids
 Math. Comp
, 2004
"... Some recovery type error estimators for linear finite element method are analyzed under O(h ) (# > 0) regular grids. Superconvergence of order O(h ) (0 < # #) is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recov ..."
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Cited by 34 (19 self)
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Some recovery type error estimators for linear finite element method are analyzed under O(h ) (# > 0) regular grids. Superconvergence of order O(h ) (0 < # #) is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.
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 ..."
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Cited by 31 (9 self)
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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 ..."
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Cited by 29 (4 self)
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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.
R.: Matlab implementation of the finite element method in elasticity
 Computing
, 2002
"... A short Matlab implementation for P1 and Q1 finite elements (FE) is provided for the numerical solution of 2d and 3d problems in linear elasticity with mixed boundary conditions. Any adaptation from the simple model examples provided to more complex problems can easily be performed with the given do ..."
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Cited by 25 (3 self)
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A short Matlab implementation for P1 and Q1 finite elements (FE) is provided for the numerical solution of 2d and 3d problems in linear elasticity with mixed boundary conditions. Any adaptation from the simple model examples provided to more complex problems can easily be performed with the given documentation. Numerical examples with postprocessing and error estimation via an averaged stress field illustrate the new Matlab tool and its flexibility.
An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints
, 2006
"... We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residualtype a posteriori error estimator that consists of e ..."
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Cited by 22 (8 self)
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We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residualtype a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.
A unifying theory of a posteriori error control for nonconforming finite element methods
 Numer. Math
"... Abstract. Residualbased a posteriori error estimates were derived within one unifying framework for lowestorder conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617637]. Therein, the key assumption is that the conforming firstorder f ..."
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Cited by 22 (8 self)
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Abstract. Residualbased a posteriori error estimates were derived within one unifying framework for lowestorder conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617637]. Therein, the key assumption is that the conforming firstorder finite element space V c h annulates the linear and bounded residual ℓ written V c h ⊆ ker ℓ. That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that V c h 6 ⊂ ker ℓ. The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator Π: V c h → V nc h with some elementary properties. It is conjectured that the more general hypothesis (H1)(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and NavierLame ́ equations illustrate the presented unifying theory of a posteriori error control for nonconforming finite element methods. 1. Unified Mixed Approach to Error Control Suppose that the primal variable u ∈ V (e.g., the displacement field) is accompanied by a dual variable p ∈ L (e.g., the flux or stress field). Typically L is some Lebesgue and V is some Sobolev space; suppose throughout this paper that L and V are Hilbert spaces and X: = L × V. Given bounded bilinear forms (1.1) a: L × L → R and b: L × V → R and well established conditions on a and b [16, 20], the linear and bounded operator
A Unifying Theory of A Posteriori Finite Element Error Control
"... Abstract. Residualbased a posteriori error estimates are derived within a unified setting for lowestorder conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm ‖ℓ ‖ of a linear functional of the form ..."
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Cited by 21 (10 self)
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Abstract. Residualbased a posteriori error estimates are derived within a unified setting for lowestorder conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm ‖ℓ ‖ of a linear functional of the form
Three Matlab implementations of the lowestorder RaviartThomas MFEM with a posteriori error control
 Computational Methods in Applied Mathematics
, 2005
"... Abstract — The numerical approximation of the Laplace equation with inhomogeneous mixed boundary conditions in 2D with lowestorder RaviartThomas mixed finite elements is realized in three flexible and short MATLAB programs. The first, hybrid, implementation (LMmfem) assumes that the discrete func ..."
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Cited by 20 (2 self)
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Abstract — The numerical approximation of the Laplace equation with inhomogeneous mixed boundary conditions in 2D with lowestorder RaviartThomas mixed finite elements is realized in three flexible and short MATLAB programs. The first, hybrid, implementation (LMmfem) assumes that the discrete function ph(x) equals a + bx for x with unknowns a ∈ R2 and b ∈ R on each element and then enforces ph ∈ H(div,Ω) through Lagrange multipliers. The second, direct, approach (EBmfem) utilizes edgebasis functions (ψE: E ∈ E) as an explicit basis of RT0 with the edgewise constant flux normal ph ·νE as a degree of freedom. The third ansatz (CRmfem) utilizes the P1 nonconforming finite element method due to Crouzeix and Raviart and then postprocesses the discrete flux via a technique due to Marini. It is the aim of this paper to derive, document, illustrate, and validate the three MATLAB implementations EBmfem, LMmfem, and CRmfem for further use and modification in education and research. A posteriori error control with a reliable and efficient averaging technique is included to monitor the discretization error. Therein, emphasis is on the correct treatment of mixed boundary conditions. Numerical examples illustrate some applications of the provided software and the quality of the error estimation.
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 17 (6 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.
Multigrid methods for obstacle problems
"... Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which ..."
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Cited by 15 (3 self)
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Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set. 1.