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A Posteriori Error Analysis and Adaptive Methods for Parabolic Problems

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Book cover Recent Progress in Computational and Applied PDES

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Abstract

We report the recent progress in deriving sharp a posteriori error estimates for linear and nonlinear parabolic problems. We show how to use special properties of a linear dual problem in non-divergence form with vanishing diffusion and strong advection to derive L 1 L 1 norm estimate for the continuous casting problem. This estimate exhibits a mild explicit dependence on velocity. We next use a direct energy estimate method to develop an efficient and reliable a posteriori error estimator for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. A convergent 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. The key ingredient in the convergence analysis is a new coarsening strategy.

Partially supported by China NSF Gram 10025102 and China National Key Project “Large Scale Scientific Computation Research” (G1999032802).

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Author information

Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100080, China

    Zhiming Chen

Authors
  1. Zhiming Chen
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Editor information

Editors and Affiliations

  1. University of California at Los Angeles, Los Angeles, California, USA

    Tony F. Chan

  2. Xiangtan University, Xiangtan City of Hunan, China

    Yunqing Huang

  3. Hong Kong Baptist University, Hong Kong, China

    Tao Tang

  4. Pennsylvania State University, University Park, Pennsylvania, USA

    Jinchao Xu

  5. Peking University, Beijing, China

    Long-An Ying

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© 2002 Springer Science+Business Media New York

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Chen, Z. (2002). A Posteriori Error Analysis and Adaptive Methods for Parabolic Problems. In: Chan, T.F., Huang, Y., Tang, T., Xu, J., Ying, LA. (eds) Recent Progress in Computational and Applied PDES. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0113-8_10

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  • DOI: https://doi.org/10.1007/978-1-4615-0113-8_10

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-4929-7

  • Online ISBN: 978-1-4615-0113-8

  • eBook Packages: Springer Book Archive

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