Numerische Mathematik

, Volume 68, Issue 2, pp 189–213

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

  • Ali Ait Ou Ammi
  • Martine Marion

DOI: 10.1007/s002110050056

Cite this article as:
Ammi, A. & Marion, M. Numer. Math. (1994) 68: 189. doi:10.1007/s002110050056

Summary.

A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces \(X_H\) and \(X_h\) for the approximation of the velocity, defined respectively on one coarse grid with grid size \(H\) and one fine grid with grid size \(h \ll H\) and one finite element space \(M_h\) for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity (\(H^1 (\Omega)\) and pressure \((L^2(\Omega)\) norm). We also discuss a penalized version of our algorithm which enjoys similar properties.

Mathematics Subject Classification (1991): 65N30 

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Ali Ait Ou Ammi
    • 1
  • Martine Marion
    • 1
  1. 1.Departement Math\'ematiques-Informatique-Syst\`emes, Ecole Centrale de Lyon, BP 163, F-69131 Ecully, France FR