This chapter illustrates the use of algebraic flux correction in the context of
finite element methods for the incompressible Navier-Stokes equations and related
models. In the convection-dominated flow regime, nonlinear stability is enforced using
algebraic flux correction. The numerical treatment of the incompressibility constraint
is based on the ‘Multilevel Pressure Schur Complement’ (MPSC) approach.
This class of iterative methods for discrete saddle-point problems unites fractionalstep
/ operator-splitting methods and strongly coupled solution techniques. The implementation
of implicit high-resolution schemes for incompressible flow problems
requires the use of efficient Newton-like methods and optimized multigrid solvers
for linear systems. The coupling of the Navier-Stokes system with scalar conservation
laws is also discussed in this chapter. The applications to be considered include
the Boussinesq model of natural convection, the k–e turbulence model, population
balance equations for disperse two-phase flows, and level set methods for free interfaces.
A brief description of the numerical algorithm is given for each problem