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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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