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Cascadic multigrid methods for parabolic problems

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Abstract

In this paper, we consider the cascadic multigrid method for a parabolic type equation. Backward Euler approximation in time and linear finite element approximation in space are employed. A stability result is established under some conditions on the smoother. Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter, these conditions are verified for a number of popular smoothers. Optimal error bounds are derived for both smooth and non-smooth data. Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.

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Authors and Affiliations

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

    Qiang Du

  2. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    PingBing Ming

Authors
  1. Qiang Du
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  2. PingBing Ming
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Correspondence to PingBing Ming.

Additional information

The work of the first author was supported in part by the National Science Foundation (Grant Nos. DMS-0409297, DMR-0205232, CCF-0430349) and US National Institute of Health-National Cancer Institute (Grant No. 1R01CA125707-01A1). The work of the second author was supported by the National Natural Science Foundation of China (Grant No. 10571172), the National Basic Research Program (Grant No. 2005CB321704), and the Youth’s Innovative Program of Chinese Academy of Sciences (Grant Nos. K7290312G9, K7502712F9).

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Du, Q., Ming, P. Cascadic multigrid methods for parabolic problems. Sci. China Ser. A-Math. 51, 1415–1439 (2008). https://doi.org/10.1007/s11425-008-0112-1

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