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On Adaptive Eulerian–Lagrangian Method for Linear Convection–Diffusion Problems

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Abstract

In this paper, we consider the adaptive Eulerian–Lagrangian method (ELM) for linear convection–diffusion problems. Unlike classical a posteriori error estimations, we estimate the temporal error along the characteristics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposed error bound, we are able to show the optimal convergence rate of ELM for solutions with minimal regularity. Furthermore, by combining this error bound with a standard residual-type estimator for the spatial error, we obtain a posteriori error estimators for a fully discrete scheme. We present numerical tests to demonstrate the efficiency and robustness of our adaptive algorithm.

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Acknowledgments

The authors would like to thank Professor Ricardo H. Nochetto, Professor Long Chen, and two anonymous referees for their comments on earlier versions of this paper. Hu and Xu are supported in part by NSF Grant DMS-0915153 and DOE Grant DE-SC0006903. Zhang is partially supported by the Dean Startup Fund, Academy of Mathematics and System Sciences, NSFC-91130011, and State High Tech Development Plan of China (863 Program) 2012AA01A309.

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Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Xiaozhe Hu & Jinchao Xu

  2. Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, China

    Xiaozhe Hu

  3. Department of the Mathematics, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA

    Young-Ju Lee

  4. NCMIS and LSEC, Academy of Mathematics and System Sciences, Beijing, 100190, China

    Chen-Song Zhang

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  1. Xiaozhe Hu
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Correspondence to Xiaozhe Hu.

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Hu, X., Lee, YJ., Xu, J. et al. On Adaptive Eulerian–Lagrangian Method for Linear Convection–Diffusion Problems. J Sci Comput 58, 90–114 (2014). https://doi.org/10.1007/s10915-013-9731-y

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  • DOI: https://doi.org/10.1007/s10915-013-9731-y

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