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Volume 30, Issue 5
Error Reduction, Convergence and Optimality for Adaptive Mixed Finite Element Methods for Diffusion Equations

Shaohong Du & Xiaoping Xie

DOI: 10.4208/jcm.1112-m3480

J. Comp. Math., 30 (2012), pp. 483-503.

Published online: 2012-10

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  • Abstract

Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods (AMFEM) for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of the stress variable error and the scaled error estimator, is shown to reduce with a fixed factor between two successive adaptive loops, up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to the quasi-error plus the divergence of the flux error. Finally, the quasi-optimal convergence rate is established for the total error, i.e. the stress variable error plus the data oscillation.

  • Keywords

Adaptive mixed finite element method, Error reduction, Convergence, Quasi-optimal convergence rate.

  • AMS Subject Headings

65N30, 65N15, 65N12, 65N50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-30-483, author = {}, title = {Error Reduction, Convergence and Optimality for Adaptive Mixed Finite Element Methods for Diffusion Equations}, journal = {Journal of Computational Mathematics}, year = {2012}, volume = {30}, number = {5}, pages = {483--503}, abstract = {

Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods (AMFEM) for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of the stress variable error and the scaled error estimator, is shown to reduce with a fixed factor between two successive adaptive loops, up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to the quasi-error plus the divergence of the flux error. Finally, the quasi-optimal convergence rate is established for the total error, i.e. the stress variable error plus the data oscillation.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1112-m3480}, url = {http://global-sci.org/intro/article_detail/jcm/8445.html} }
TY - JOUR T1 - Error Reduction, Convergence and Optimality for Adaptive Mixed Finite Element Methods for Diffusion Equations JO - Journal of Computational Mathematics VL - 5 SP - 483 EP - 503 PY - 2012 DA - 2012/10 SN - 30 DO - http://doi.org/10.4208/jcm.1112-m3480 UR - https://global-sci.org/intro/article_detail/jcm/8445.html KW - Adaptive mixed finite element method, Error reduction, Convergence, Quasi-optimal convergence rate. AB -

Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods (AMFEM) for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of the stress variable error and the scaled error estimator, is shown to reduce with a fixed factor between two successive adaptive loops, up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to the quasi-error plus the divergence of the flux error. Finally, the quasi-optimal convergence rate is established for the total error, i.e. the stress variable error plus the data oscillation.

Shaohong Du & Xiaoping Xie. (1970). Error Reduction, Convergence and Optimality for Adaptive Mixed Finite Element Methods for Diffusion Equations. Journal of Computational Mathematics. 30 (5). 483-503. doi:10.4208/jcm.1112-m3480
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