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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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  • TXT
@Article{JCM-30-483, author = {Shaohong Du and Xiaoping Xie}, 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 AU - Shaohong Du & Xiaoping Xie 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 and Xiaoping Xie. (2012). 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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