  Convection-diffusion-reaction equations model the conservation of scalar
quantities. From the analytic point of view, solution of these equations
satisfy under certain conditions maximum principles, which represent physical
bounds of the solution. That the same bounds are respected by numerical
approximations of the solution is often of utmost importance in practice. The
mathematical formulation of this property, which contributes to the physical
consistency of a method, is called Discrete Maximum Principle (DMP). In many
applications, convection dominates diffusion by several orders of magnitude. It
is well known that standard discretizations typically do not satisfy the DMP in
this convection-dominated regime. In fact, in this case, it turns out to be a
challenging problem to construct discretizations that, on the one hand, respect
the DMP and, on the other hand, compute accurate solutions. This paper presents
a survey on finite element methods, with a main focus on the
convection-dominated regime, that satisfy a local or a global DMP. The concepts
of the underlying numerical analysis are discussed. The survey reveals that for
the steady-state problem there are only a few discretizations, all of them
nonlinear, that at the same time satisfy the DMP and compute reasonably
accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of
these discretizations have been developed in recent years, showing the enormous
progress that has been achieved lately. Methods based on algebraic
stabilization, nonlinear and linear ones, are currently as well the only finite
element methods that combine the satisfaction of the global DMP and accurate
numerical results for the evolutionary equations in the convection-dominated
situation.
