Error Reduction in Adaptive Finite Element Approximations of Elliptic Obstacle Problems

  • We consider an adaptive finite element method (AFEM) for obstacle problems associated with linear second order elliptic boundary value problems and prove a reduction in the energy norm of the discretization error which leads to R-linear convergence. This result is shown to hold up to a consistency error due to the extension of the discrete multipliers (point functionals) to H^{-1} and a possible mismatch between the continuous and discrete coincidence and noncoincidence sets. The AFEM is based on a residual-type error estimator consisting of element and edge residuals. The a posteriori error analysis reveals that the significant difference to the unconstrained case lies in the fact that these residuals only have to be taken into account within the discrete noncoincidence set. The proof of the error reduction property uses the reliability and the discrete local efficiency of the estimator as well as a perturbed Galerkin orthogonality. Numerical results are given illustrating theWe consider an adaptive finite element method (AFEM) for obstacle problems associated with linear second order elliptic boundary value problems and prove a reduction in the energy norm of the discretization error which leads to R-linear convergence. This result is shown to hold up to a consistency error due to the extension of the discrete multipliers (point functionals) to H^{-1} and a possible mismatch between the continuous and discrete coincidence and noncoincidence sets. The AFEM is based on a residual-type error estimator consisting of element and edge residuals. The a posteriori error analysis reveals that the significant difference to the unconstrained case lies in the fact that these residuals only have to be taken into account within the discrete noncoincidence set. The proof of the error reduction property uses the reliability and the discrete local efficiency of the estimator as well as a perturbed Galerkin orthogonality. Numerical results are given illustrating the performance of the AFEM.show moreshow less

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Metadaten
Author:Dietrich BraessGND, Carsten CarstensenGND, Ronald H. W. HoppeGND
URN:urn:nbn:de:bvb:384-opus4-5153
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/653
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2008-18)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Contributing Corporation:Ruhr-Universitaet Bochum, Humboldt Universitaet zu Berlin, University of Houston
Release Date:2008/04/15
Tag:adaptive finite element methods; elliptic obstacle problems; error reduction
GND-Keyword:Finite-Elemente-Methode; Hindernisproblem; Elliptisches Randwertproblem; Fehleranalyse; A-posteriori-Abschätzung
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik