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Local and parallel algorithms for fourth order problems discretized by the Morley–Wang–Xu element method

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Abstract

This paper systematically studies numerical solution of fourth order problems in any dimensions by use of the Morley–Wang–Xu (MWX) element discretization combined with two-grid methods (Xu and Zhou (Math Comp 69:881–909, 1999)). Since the coarse and fine finite element spaces are nonnested, two intergrid transfer operators are first constructed in any dimensions technically, based on which two classes of local and parallel algorithms are then devised for solving such problems. Following some ideas in (Xu and Zhou (Math Comp 69:881–909, 1999)), the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems (Huang et al. (Appl Numer Math 37:519–533, 2001); Huang et al. (Sci China Ser A 49:109–120, 2006)), and the error estimates for the intergrid transfer operators, we prove that the discrete energy errors of the two classes of methods are of the sizes O(h + H 2) and O(h + H 2(H/h)(d−1)/2), respectively. Here, H and h denote respectively the mesh sizes of the coarse and fine finite element triangulations, and d indicates the space dimension of the solution region. Numerical results are performed to support the theory obtained and to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Jianguo Huang & Xuehai Huang

  2. Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai, China

    Jianguo Huang

  3. College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035, China

    Xuehai Huang

Authors
  1. Jianguo Huang
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  2. Xuehai Huang
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Correspondence to Jianguo Huang or Xuehai Huang.

Additional information

The work of the first author was partially supported by the NNSFC (11161130004) and E-Institutes of Shanghai Municipal Education Commission (E03004). The work of the second author was partially supported by the Zhejiang Provincial Natural Science Foundation of China (Y6110240).

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Huang, J., Huang, X. Local and parallel algorithms for fourth order problems discretized by the Morley–Wang–Xu element method. Numer. Math. 119, 667–697 (2011). https://doi.org/10.1007/s00211-011-0396-x

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  • DOI: https://doi.org/10.1007/s00211-011-0396-x

Mathematics Subject Classification (2000)