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An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems


Authors: Zhiming Chen and Jia Feng
Translated by:
Journal: Math. Comp. 73 (2004), 1167-1193
MSC (2000): Primary 65N15, 65N30, 65N50
DOI: https://doi.org/10.1090/S0025-5718-04-01634-5
Published electronically: January 23, 2004
MathSciNet review: 2047083
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Abstract: An efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. It is proved that at each time step the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within a finite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm.


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Additional Information

Zhiming Chen
Affiliation: LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China
Email: zmchen@lsec.cc.ac.cn

Jia Feng
Affiliation: Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, Peoples Republic of China
Email: fjia@lsec.cc.ac.cn

DOI: https://doi.org/10.1090/S0025-5718-04-01634-5
Received by editor(s): September 28, 2001
Received by editor(s) in revised form: January 12, 2003
Published electronically: January 23, 2004
Additional Notes: The first author was supported in part by China NSF under the grant 10025102 and by China MOS under the grant G1999032802
Article copyright: © Copyright 2004 American Mathematical Society

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