Volume 27, Issue 2-3
Error Reduction in Adaptive Finite Element Approximations of Elliptic Obstacle Problems

Dietrich Braess, Carsten Carstensen & Ronald H.W. Hoppe

DOI:

J. Comp. Math., 27 (2009), pp. 148-169.

Published online: 2009-04

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  • Abstract

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 the performance of the AFEM.

  • Keywords

Adaptive finite element methods Elliptic obstacle problems Convergence analysis

  • AMS Subject Headings

65N30 65N50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-148, author = {}, title = {Error Reduction in Adaptive Finite Element Approximations of Elliptic Obstacle Problems}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {2-3}, pages = {148--169}, abstract = {

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 the performance of the AFEM.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8565.html} }
TY - JOUR T1 - Error Reduction in Adaptive Finite Element Approximations of Elliptic Obstacle Problems JO - Journal of Computational Mathematics VL - 2-3 SP - 148 EP - 169 PY - 2009 DA - 2009/04 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8565.html KW - Adaptive finite element methods KW - Elliptic obstacle problems KW - Convergence analysis AB -

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 the performance of the AFEM.

Dietrich Braess, Carsten Carstensen & Ronald H.W. Hoppe. (2019). Error Reduction in Adaptive Finite Element Approximations of Elliptic Obstacle Problems. Journal of Computational Mathematics. 27 (2-3). 148-169. doi:
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