This chapter is concerned with the design of high-resolution finite element
schemes satisfying the discrete maximum principle. The presented algebraic
flux correction paradigm is a generalization of the flux-corrected transport (FCT)
methodology. Given the standard Galerkin discretization of a scalar transport equation,
we decompose the antidiffusive part of the discrete operator into numerical
fluxes and limit these fluxes in a conservative way. The purpose of this manipulation
is to make the antidiffusive term local extremum diminishing. The available limiting
techniques include a family of implicit FCT schemes and a new linearity-preserving
limiter which provides a unified treatment of stationary and time-dependent problems.
The use of Anderson acceleration makes it possible to design a simple and
efficient quasi-Newton solver for the constrained Galerkin scheme. We also present
a linearized FCT method for computations with small time steps. The numerical
behavior of the proposed algorithms is illustrated by a grid convergence study for
convection-dominated transport problems and anisotropic diffusion equations.