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Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization

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Abstract

A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the lowest-dimensional space \(H_{m_{1}}\) with the largest time step Δt 1; subsequent approximations are generated on a succession of higher-dimensional spaces \(H_{m_{j}}\) with small time step Δt j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal accuracy of the approximate solution in H 1-norm and L 2-norm are investigated, i.e., m j∼m 3/2j−1 , Δt j∼Δt 3/2j−1 , j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method.

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Authors and Affiliations

  1. Faculty of Science (State Key Laboratory of Multiphase Flow in Power Engineering), Xi'an Jiaotong University, Xi'an 710049, P. R. China

    Yinnian He

  2. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

    Kam-Moon Liu

Authors
  1. Yinnian He
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  2. Kam-Moon Liu
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Correspondence to Yinnian He.

Additional information

Communicated by J. Xu

Mathematics subject classifications (2000)

35L70, 65N30, 76D06

Subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China 10371095 and the City University of Hong Kong Research Project 7001093, NSF of China 50323001.

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He, Y., Liu, KM. Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization. Adv Comput Math 25, 403–433 (2006). https://doi.org/10.1007/s10444-004-7640-1

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  • DOI: https://doi.org/10.1007/s10444-004-7640-1

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