E. C. Cyr, J. N. Shadid, R. S. Tuminaro,
Stabilization and Scalable Block Preconditioning for the Navier-Stokes
Equations, J. Comp. Phys. 231:345-363, 2012. (SAND Number 2011-6456 J)
This study compares several block-oriented preconditioners for the stabilized
finite element discretization of the incompressible Navier-Stokes equations.
This includes standard additive Schwarz domain decomposition methods, aggressive
coarsening multigrid, and three preconditioners based on an approximate block LU
factorization, specifically SIMPLEC, LSC, and PCD. Robustness is considered with
a particular focus on the impact that different stabilization methods have on
preconditioner performance. Additionally, parallel scaling studies are undertaken.
The numerical results indicate that aggressive coarsening multigrid, LSC and PCD all have
good algorithmic scalability. Coupling this with the fact that block methods can
be applied to systems arising from stable mixed discretizations implies that these
techniques are a promising direction for developing scalable methods for Navier-Stokes.