This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block factorization of the original system matrix. We have considered a wide set of methods: algebraic pressure correction methods, algebraic velocity correction methods and the Yosida method. Higher order schemes, based on improved factorizations, are also introduced. We have also explained the relationship between these pressure segregation methods and some widely used preconditioners, and we have introduced predictor-corrector methods, one-loop algorithms where nonlinearity and iterations towards the monolithic system are coupled.

L. Yang, J. Yang, E. Boek, M. Sakai, C. Pain. Image-based simulations of absolute permeability with massively parallel pseudo-compressible stabilised finite element solver. Comput Geosci 23(5) (2019) DOI 10.1007/s10596-019-09837-4

A. Pont, R. Codina, J. Baiges. Interpolation with restrictions between finite element meshes for flow problems in an ALE setting. Int. J. Numer. Meth. Engng 110(13) (2016) DOI 10.1002/nme.5444

Y. Notay. Convergence of Some Iterative Methods for Symmetric Saddle Point Linear Systems. SIAM J. Matrix Anal. Appl. 40(1) (2019) DOI 10.1137/18m1208836

E. Ferrer, D. Moxey, R. Willden, S. Sherwin. Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations. Commun. comput. phys. 16(3) (2015) DOI 10.4208/cicp.290114.170414a

R. Codina. On the Design of Algebraic Fractional Step Methods for Viscoelastic Incompressible Flows. (2018) DOI 10.1007/978-3-319-97613-6_6