Volume 32, Issue 5
On Residual-Based a Posteriori Error Estimators for Lowest-Order Raviart-Thomas Element Approximation to Convection-Diffusion-Reaction Equations

J. Comp. Math., 32 (2014), pp. 522-546.

Published online: 2014-10

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• Abstract

A new technique of residual-type a posteriori error analysis is developed for the lowest-order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations in two- or three-dimension. Both centered mixed scheme and upwind-weighted mixed scheme are considered. The a posteriori error estimators, derived for the stress variable error plus scalar displacement error in $L^2$-norm, can be directly computed with the solutions of the mixed schemes without any additional cost, and are proven to be reliable. Local efficiency dependent on local variations in coefficients is obtained without any saturation assumption, and holds from the cases where convection or reaction is not present to convection- or reaction-dominated problems. The main tools of the analysis are the postprocessed approximation of scalar displacement, abstract error estimates, and the property of modified Oswald interpolation. Numerical experiments are carried out to support our theoretical results and to show the competitive behavior of the proposed posteriori error estimates.

• Keywords

Convection-diffusion-reaction equation, Centered mixed scheme, Upwind-weighted mixed scheme, Postprocessed approximation, A posteriori error estimators.

65N15, 65N30, 76S05.

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@Article{JCM-32-522, author = {}, title = {On Residual-Based a Posteriori Error Estimators for Lowest-Order Raviart-Thomas Element Approximation to Convection-Diffusion-Reaction Equations}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {5}, pages = {522--546}, abstract = {

A new technique of residual-type a posteriori error analysis is developed for the lowest-order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations in two- or three-dimension. Both centered mixed scheme and upwind-weighted mixed scheme are considered. The a posteriori error estimators, derived for the stress variable error plus scalar displacement error in $L^2$-norm, can be directly computed with the solutions of the mixed schemes without any additional cost, and are proven to be reliable. Local efficiency dependent on local variations in coefficients is obtained without any saturation assumption, and holds from the cases where convection or reaction is not present to convection- or reaction-dominated problems. The main tools of the analysis are the postprocessed approximation of scalar displacement, abstract error estimates, and the property of modified Oswald interpolation. Numerical experiments are carried out to support our theoretical results and to show the competitive behavior of the proposed posteriori error estimates.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1403-FE4}, url = {http://global-sci.org/intro/article_detail/jcm/9902.html} }
TY - JOUR T1 - On Residual-Based a Posteriori Error Estimators for Lowest-Order Raviart-Thomas Element Approximation to Convection-Diffusion-Reaction Equations JO - Journal of Computational Mathematics VL - 5 SP - 522 EP - 546 PY - 2014 DA - 2014/10 SN - 32 DO - http://doi.org/10.4208/jcm.1403-FE4 UR - https://global-sci.org/intro/article_detail/jcm/9902.html KW - Convection-diffusion-reaction equation, Centered mixed scheme, Upwind-weighted mixed scheme, Postprocessed approximation, A posteriori error estimators. AB -

A new technique of residual-type a posteriori error analysis is developed for the lowest-order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations in two- or three-dimension. Both centered mixed scheme and upwind-weighted mixed scheme are considered. The a posteriori error estimators, derived for the stress variable error plus scalar displacement error in $L^2$-norm, can be directly computed with the solutions of the mixed schemes without any additional cost, and are proven to be reliable. Local efficiency dependent on local variations in coefficients is obtained without any saturation assumption, and holds from the cases where convection or reaction is not present to convection- or reaction-dominated problems. The main tools of the analysis are the postprocessed approximation of scalar displacement, abstract error estimates, and the property of modified Oswald interpolation. Numerical experiments are carried out to support our theoretical results and to show the competitive behavior of the proposed posteriori error estimates.

Shaohong Du & Xiaoping Xie. (1970). On Residual-Based a Posteriori Error Estimators for Lowest-Order Raviart-Thomas Element Approximation to Convection-Diffusion-Reaction Equations. Journal of Computational Mathematics. 32 (5). 522-546. doi:10.4208/jcm.1403-FE4
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