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Convergence of adaptive finite element methods
 SIAM Review
"... Abstract. We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasiorthogonality, which accounts for the bilinear form not being a scalar product, together ..."
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Cited by 72 (6 self)
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Abstract. We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasiorthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEM is a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and convectiondiffusion PDE, illustrate the theory and yield optimal meshes.
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.
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 ..."
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Cited by 20 (6 self)
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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.
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.
On the convergence of adaptive nonconforming finite element methods for a class of convex variational problems
 SIAM J. Numer. Anal
"... tion. Part I: the Full Space Problem ..."
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Convergence of adaptive edge element methods for the 3D eddy currents equations
 J. Comp. Math
"... Abstract. We consider an Adaptive Edge Finite Element Method (AEFEM) for the 3D eddy currents equations with variable coefficients using a residualtype a posteriori error estimator. Both the components of the estimator and certain oscillation terms, due to the occurrence of the variable coefficient ..."
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Cited by 6 (3 self)
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Abstract. We consider an Adaptive Edge Finite Element Method (AEFEM) for the 3D eddy currents equations with variable coefficients using a residualtype a posteriori error estimator. Both the components of the estimator and certain oscillation terms, due to the occurrence of the variable coefficients, have to be controlled properly within the adaptive loop which is taken care of by appropriate bulk criteria. Convergence of the AEFEM in terms of reductions of the energy norm of the discretization error and of the oscillations is shown. Numerical results are given to illustrate the performance of the AEFEM. Key words. adaptive edge elements, 3D eddy currents equations, convergence analysis, error and oscillation reduction, residual type a posteriori error estimates AMS subject classifications. 65F10, 65N30 1. Introduction. The
Convergence of an adaptive finite element method on quadrilateral meshes
, 2008
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Functional approach to a posteriori error estimation for elliptic optimal control problems with distributed control
 JOURNAL OF MATHEMATICAL SCIENCES
, 2007
"... We present a new approach to the a posteriori analysis of distributed optimal control problems. It is based on functional type a posteriori estimates that provide computable and guaranteed bounds of errors for any conforming approximations of a boundary value problem. We derive computable twosid ..."
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Cited by 4 (4 self)
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We present a new approach to the a posteriori analysis of distributed optimal control problems. It is based on functional type a posteriori estimates that provide computable and guaranteed bounds of errors for any conforming approximations of a boundary value problem. We derive computable twosided a posteriori estimates for the cost functional and estimates for the approximations of state and control functions. Numerical results illustrate efficiency of the approach suggested.
Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem
 J. Sci. Comput
"... ar ..."
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