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A Parallel Partition of Unity Scheme Based on Two-Grid Discretizations for the Navier–Stokes Problem

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Abstract

A parallel partition of unity scheme based on two-grid discretizations is proposed and analyzed in this paper for the Navier–Stokes problem. A standard Galerkin finite element method on a relatively coarse grid is used to obtain the approximation of the lower frequency components and the higher frequency components are computed on fine grids by some local and parallel procedure. The motivation of the proposed parallel partition of unity scheme is based on the superposition principle. That is the original linear problem, which is a linear residual equation, can be equivalent to the sum of a series of simple problems of same type with free terms of small supports. Each simple problem is approximated by a local problem with homogeneous Dirichlet boundary condition. Optimal error estimates are obtained and some numerical tests are presented to support the theoretical results.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250014, China

    Guangzhi Du

  2. School of Mathematical Sciences, University of Jinan, Jinan, Shandong, 250022, China

    Liyun Zuo

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  1. Guangzhi Du
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  2. Liyun Zuo
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Correspondence to Guangzhi Du.

Additional information

G. Du: Subsidized by NSFC (Grant No. 11701343) and partially supported by NSFC (Grant Nos. 11571274, 11401466)

L. Zuo: Subsidized by the Provincial Natural Science Foundation of Shandong (Grant No. ZR2017BA027).

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Du, G., Zuo, L. A Parallel Partition of Unity Scheme Based on Two-Grid Discretizations for the Navier–Stokes Problem. J Sci Comput 75, 1445–1462 (2018). https://doi.org/10.1007/s10915-017-0593-6

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  • DOI: https://doi.org/10.1007/s10915-017-0593-6

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