Isogeometric residual minimization (iGRM) for non-stationary Stokes and Navier–Stokes problems

M. Łoś, IGNACIO PATRICIO PEDRO MUGA URQUIZA, J. Muñoz-Matute, M. Paszyński

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We show that it is possible to obtain a linear computational cost FEM-based solver for non-stationary Stokes and Navier–Stokes equations. Our method employs a technique developed by Guermond and Minev (2011), which consists of singular perturbation plus a splitting scheme. While the time-integration schemes are implicit, we use finite elements to discretize the spatial counterparts. At each time-step, we solve a PDE having weak-derivatives in one direction only (which allows for the linear computational cost), at the expense of handling strong second-order derivatives of the previous time step solution, on the right-hand side of these PDEs. This motivates the use of smooth functions such as B-splines. For high Reynolds numbers, some of these PDEs become unstable. To deal robustly with these instabilities, we propose to use a residual minimization technique. We test our method on problems having manufactured solutions, as well as on the cavity flow problem.

Original languageEnglish
JournalComputers and Mathematics with Applications
DOIs
StateAccepted/In press - 2020
Externally publishedYes

Keywords

  • Alternating directions
  • Isogeometric analysis
  • Linear computational cost solver
  • Navier–Stokes problem
  • Non-stationary Stokes
  • Residual minimization

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