Algebraic flux correction III. Incompressible flow problems

Abstract 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 us-ing algebraic flux correction. The numerical treatment of the incompressibility con-straint is based on the ‘Multilevel Pressure Schur Complement ’ (MPSC) approach. This class of iterative methods for discrete saddle-point problems unites fractional-step / operator-splitting methods and strongly coupled solution techniques. The im-plementation 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 conserva-tion laws is also discussed in this chapter. The applications to be considered include the Boussinesq model of natural convection, the k–ε turbulence model, population balance equations for disperse two-phase flows, and level set methods for free inter-faces. A brief description of the numerical algorithm is given for each problem.