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Analysis of a projection method for the Stokes problem using an \(\varepsilon \)-Stokes approach

  • Masato Kimura
  • Kazunori MatsuiEmail author
  • Adrian Muntean
  • Hirofumi Notsu
Original Paper
  • 23 Downloads

Abstract

We generalize pressure boundary conditions of an \(\varepsilon \)-Stokes problem. Our \(\varepsilon \)-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter \(\varepsilon >0\). For the Dirichlet boundary condition, it is proven in Matsui and Muntean (Adv Math Sci Appl, 27:181–191, 2018) that the solution for the \(\varepsilon \)-Stokes problem converges to the one for the Stokes problem as \(\varepsilon \) tends to 0, and to the one for the pressure-Poisson problem as \(\varepsilon \) tends to \(\infty \). Here, we extend these results to the Neumann and mixed boundary conditions. We also establish error estimates in suitable norms between the solutions to the \(\varepsilon \)-Stokes problem, the pressure-Poisson problem and the Stokes problem, respectively. Several numerical examples are provided to show that several such error estimates are optimal in \(\varepsilon \). Our error estimates are improved if one uses the Neumann boundary conditions. In addition, we show that the solution to the \(\varepsilon \)-Stokes problem has a nice asymptotic structure.

Keywords

Stokes problem Pressure-Poisson equation Asymptotic analysis Finite element method 

Mathematics Subject Classification

76D03 35Q35 35B40 65N30 

Notes

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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsKanazawa UniversityKanazawaJapan
  2. 2.Division of Mathematical and Physical Sciences, Graduate School of Natural Science and TechnologyKanazawa UniversityKanazawaJapan
  3. 3.Department of Mathematics and Computer ScienceKarlstad UniversityKarlstadSweden
  4. 4.Japan Science and Technology Agency, PRESTOKawaguchiJapan

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