Applications of Mathematics

, Volume 62, Issue 4, pp 377–403

A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element approximation



We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization of the penalty term, we propose reduced and non-reduced integration schemes, and obtain an error estimate for velocity and pressure. The theoretical results are verified by numerical experiments.


penalty method Stokes problem finite element method error estimate 

MSC 2010

65N30 35Q30 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  • Guanyu Zhou
    • 1
  • Takahito Kashiwabara
    • 2
  • Issei Oikawa
    • 3
  1. 1.Department of Applied MathematicsThe Tokyo University of ScienceTokyoJapan
  2. 2.The Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  3. 3.Faculty of Science and EngineeringWaseda UniversityTokyoJapan

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