Journal of Scientific Computing

, Volume 68, Issue 1, pp 339–374 | Cite as

Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition

  • Guanyu ZhouEmail author
  • Takahito Kashiwabara
  • Issei Oikawa


We consider the penalty method for the stationary Navier–Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate \(O(\epsilon )\) in \(H^k\)-norm, where \(\epsilon \) is the penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate \(O(h+\sqrt{\epsilon }+h/\sqrt{\epsilon })\) for the non-reduced-integration scheme with \(d=2,3\), and the reduced-integration scheme with \(d=3\), where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with \(d=2\), we prove the convergence order \(O(h+\sqrt{\epsilon }+h^2/\sqrt{\epsilon })\). The theoretical results are verified by numerical experiments.


Finite element method The Navier–Stokes equations  Slip boundary condition Penalty method 

Mathematics Subject Classification

65N60 35Q30 



The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  2. 2.Graduate School of Environmental and Life ScienceOkayama UniversityOkayamaJapan
  3. 3.Faculty of Science and EngineeringWaseda UniversityTokyoJapan

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