Journal of Scientific Computing

, Volume 58, Issue 1, pp 249–274 | Cite as

Spectral Method for Navier–Stokes Equations with Slip Boundary Conditions

  • Ben-yu Guo
  • Yu-jian Jiao
Original Research


In this paper, we propose a spectral method for the \(n\)-dimensional Navier–Stokes equations with slip boundary conditions by using divergence-free base functions. The numerical solutions fulfill the incompressibility and the physical boundary conditions automatically. Therefore, we need neither the artificial compressibility method nor the projection method. Moreover, we only have to evaluate the unknown coefficients of expansions of \(n-1\) components of the velocity. These facts simplify actual computation and numerical analysis essentially, and also save computational time. As the mathematical foundation of this new approach, we establish some approximation results, with which we prove the spectral accuracy in space of the proposed algorithm. Numerical results demonstrate its high efficiency and coincide the analysis very well. The main idea, the approximation results and the techniques developed in this paper are also applicable to numerical simulations of other problems with divergence-free solutions, such as certain partial differential equations describing electro-magnetic fields.


Spectral method Navier–Stokes equations Slip boundary conditions 

Mathematics Subject Classification (2000)

65M70 41A10 76D05 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Scientific Computing Key Laboratory of Shanghai UniversitiesShanghai Normal UniversityShanghaiChina
  3. 3.E-Institute for Computational Science of Shanghai UniversitiesShanghai Normal UniversityShanghaiChina

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