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146
An optimal control approach to a posteriori error estimation in finite element methods
, 2001
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GoalOriented Error Estimation and Adaptivity for the Finite Element Method
 COMPUT. MATH. APPL
, 1999
"... In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These socalled quantities of interest are characterized by linear functionals ..."
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Cited by 76 (8 self)
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In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These socalled quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundaryvalue problems, and in particular, show how to obtain accurate estimates as well as upper and lowerbounds on the error. We also study the new concept of goaloriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.
A finite element method for domain decomposition with nonmatching grids
 ESAIM: Mathematical Modeling and Numerical Analysis
"... Abstract. In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to selfadjoint elliptic problems, using Poisson’s equation a ..."
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Cited by 71 (11 self)
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Abstract. In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to selfadjoint elliptic problems, using Poisson’s equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.
A Posteriori Finite Element Bounds for LinearFunctional Outputs of Elliptic Partial Differential Equations
 Computer Methods in Applied Mechanics and Engineering
, 1997
"... We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order elliptic linear partial differential equations in two space dimensions. The method is base ..."
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Cited by 64 (9 self)
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We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order elliptic linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic "energy" reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine "truthmesh" discretization are then derived by appealing to a dual maxmin relaxation evaluated for optimally chosen adjoint and hybridflux candidate Lagrange multipliers generated by a Kelement coarser "workingmesh" approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomainlocal, symmetric Neumann pro...
Adaptive Finite Element Methods For Optimal Control Of Partial Differential Equations: Basic Concept
 SIAM J. Contr. Optim
, 1998
"... A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations. The Lagrangian formalism yields the firstorder necessary optimality condition in form of an indefinite boundary value problem whic ..."
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Cited by 61 (5 self)
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A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations. The Lagrangian formalism yields the firstorder necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive Galerkin finite element method. The mesh design in the resulting reduced models is controlled by residualbased a posteriori error estimates. These are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error. In this case, the computed state and costate variables can be used as sensitivity factors multiplying the local cellresiduals in the error estimators. This results in a generic and simple algorithm for mesh adaptation within the optimization process. This method is developed and tested for simple boundary control problems in semiconductor models.
Quasi interpolation and a posteriori error analysis in ®nite element method.
 M 2AN
, 1999
"... Abstract. One of the main tools in the proof of residualbased a posteriori error estimates is a quasiinterpolation operator due to Clément. We modify this operator in the setting of a partition of unity with the effect that the approximation error has a local average zero. This results in a new re ..."
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Cited by 46 (12 self)
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Abstract. One of the main tools in the proof of residualbased a posteriori error estimates is a quasiinterpolation operator due to Clément. We modify this operator in the setting of a partition of unity with the effect that the approximation error has a local average zero. This results in a new residualbased a posteriori error estimate with a volume contribution which is smaller than in the standard estimate. For an elliptic model problem, we discuss applications to conforming, nonconforming and mixed finite element methods.
An Adaptive Finite Element Method for Large Scale Image Processing
 INTERNATIONAL CONFERENCE ON SCALESPACE THEORIES IN COMPUTER VISION
, 1999
"... Nonlinear diffusion methods have proved to be powerful methods in the processing of 2D and 3D images. They allow a denoising and smoothing of image intensities while retaining and enhancing edges. As time evolves in the corresponding process, a scale of successively coarser image details is generate ..."
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Cited by 45 (18 self)
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Nonlinear diffusion methods have proved to be powerful methods in the processing of 2D and 3D images. They allow a denoising and smoothing of image intensities while retaining and enhancing edges. As time evolves in the corresponding process, a scale of successively coarser image details is generated. Certain features, however, remain highly resolved and sharp. On the other hand, compression is an important topic in image processing as well. Here a method is presented which combines the two aspects in an efficient way. It is based on a semi–implicit Finite Element implementation of nonlinear diffusion. Error indicators guide a successive coarsening process. This leads to locally coarse grids in areas of resulting smooth image intensity, while enhanced edges are still resolved on fine grid levels. Special emphasis has been put on algorithmical aspects such as storage requirements and efficiency. Furthermore, a new nonlinear anisotropic diffusion method for vector field visualization is presented.
Bounds for LinearFunctional Outputs of Coercive Partial Differential Equations: Local Indicators and Adaptive Refinement
, 1997
"... this paper we focus on three new developments. First, we introduce a modified energy objective, and hence modified Lagrangian, that permits both more transparent interpretation and more ready generalization. Second, we demonstrate that the bound gap  the difference between the upper and lower bou ..."
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Cited by 44 (9 self)
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this paper we focus on three new developments. First, we introduce a modified energy objective, and hence modified Lagrangian, that permits both more transparent interpretation and more ready generalization. Second, we demonstrate that the bound gap  the difference between the upper and lower bounds for the desired output  can be represented as the sum of positive contributions  local indicators  associated with the elements TH of TH . Third, based on these local boundgap error indicators, we develop adaptive strategies by which to reduce the bound gap  and hence improve our validated prediction for the output of interest  through optimal refinement of TH . The resulting method is applied to an illustrative problem in linear elasticity.
A posteriori error estimation techniques in practical finite element analysis
, 2005
"... In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goaloriented error estimates. While we show how these er ..."
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Cited by 40 (5 self)
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In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goaloriented error estimates. While we show how these error estimation techniques are employed for our simple model problem, the emphasis of the paper is on assessing whether these procedures are ready for use in practical linear finite element analysis. We conclude that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care. The more accurate estimation procedures also do not provide proven bounds that, in general, can be computed efficiently. We also briefly comment upon the state of error estimations
A posteriori error estimates for vertex centered finite volume approximations of convectiondiffusionreaction equations
 M2AN Math. Model. Numer. Anal
, 2000
"... This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convectiondiffusionreaction equation c t +r (uf(c)) r (Drc)+c = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L¹ ..."
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Cited by 39 (7 self)
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This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convectiondiffusionreaction equation c t +r (uf(c)) r (Drc)+c = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L¹norm, independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical