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Uniform optimal-order estimates for finite element methods for advection-diffusion equations

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Abstract

This article summarizes our recent work on uniform error estimates for various finite element methods for time-dependent advection-diffusion equations.

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References

  1. M. Křížek, P. Neittaanmäki, and R. Stenberg, Finite Element Methods: Superconvergence, Post-Processing and a Posteriori Estimates, Marcel Dekker, New York, 1997.

    Google Scholar 

  2. G. Strang, Calculus, Wellesley-Cambridge Press, Wellesley MA, 1991.

    Google Scholar 

  3. T. Arbogast and M. Wheeler, A characteristic-mixed finite element method, SIAM Numer. Anal., 1995, 32: 404–424.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Douglas, Jr., C. Huang, and F. Pereira, The modified method of characteristics with adjusted advection, Numer. Math., 1999, 83: 353–369.

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations (second edition), Springer-Verlag, Berlin, 2008.

    MATH  Google Scholar 

  6. H. Wang, An optimal-order error estimate for an ELLAM scheme for two-dimensional linear advection-diffusion equation, SIAM J. Numer. Anal., 2000, 37: 1338–1368.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Wang, R. E. Ewing, and T. Russell, Eulerian-Lagrangian localized methods for convection-diffusion equations and their convergence analysis, IMA J. Numer. Anal., 1995, 15: 405–459.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Bause and P. Knabner, Uniform error analysis for Lagrange-Galerkin approximation of convection dominated problems, SIAM J. Numer. Anal, 2002, 39: 1954–1984.

    Article  MATH  MathSciNet  Google Scholar 

  9. H. Wang and K. Wang, Uniform estimates for Eulerian-Lagrangian methods for singularity perturbed time-dependent problems, SIAM J. Numer. Anal., 2007, 45: 1305–1329.

    Article  MATH  MathSciNet  Google Scholar 

  10. Q. Lin, An integral identity and interpolated postprocess in superconvergence, Research Report, Institute of Systems Science, Academia Sinica, 1990, 90–97.

  11. Q. Lin and N. Yan, Construction and Analysis for Effective Finite Element Methods, Hebei University Press, Baoding, 1996.

    Google Scholar 

  12. Q. Lin, L. Tobiska, and A. Zhou, Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation, IMA J. Numer. Anal., 2005, 25: 160–181.

    Article  MATH  MathSciNet  Google Scholar 

  13. Q. Lin and J. Lin, Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, 2006.

    Google Scholar 

  14. C. Grossmann, H. Roos, and M. Stynes, Numerical Treatment of Partial Differential Equations, Springer, Berlin Heidelberg, 2007.

    Book  MATH  Google Scholar 

  15. Q. Lin, J. Zhou, and H. Chen, Superclose and extrapolation of the tetrahedral linear finite elements for poisson equation in three-dimensional rectangular field, Mathematics in Practice and Theory, 2009, 39(13): 215–220.

    Google Scholar 

  16. L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, AMS, Rhode Island, Vol. 19, 1998.

    MATH  Google Scholar 

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Correspondence to Qun Lin.

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This work was supported in part by the National Basic Research Program (2007CB814906), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300), and the National Science Foundation under Grant No. EAR-0934747.

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Lin, Q., Wang, H. & Zhang, S. Uniform optimal-order estimates for finite element methods for advection-diffusion equations. J Syst Sci Complex 22, 555–559 (2009). https://doi.org/10.1007/s11424-009-9187-1

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  • DOI: https://doi.org/10.1007/s11424-009-9187-1

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