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.

翻译：对流-扩散-反应方程用于标量守恒量的建模。从解析角度看，这些方程的解在特定条件下满足最大值原理，从而表征解的物理边界。实践中，确保数值近似解同样遵守这些边界条件往往至关重要。这一性质的数学表述称为离散最大原理（DMP），它体现了方法的物理一致性。在许多应用中，对流项的强度比扩散项高出数个数量级。众所周知，标准离散格式在此类对流主导的背景下通常不满足DMP。实际上，在此情形下，构造既满足DMP又能精确计算解的离散格式极具挑战性。本文综述了满足局部或全局DMP的有限元方法，重点关注对流主导情形。文中讨论了相关数值分析的核心概念。综述表明，对于稳态问题，仅有少数离散格式（均为非线性）能同时满足DMP并生成足够精确的解，例如代数稳定化格式。值得注意的是，此类格式大多在近年发展起来，体现了该领域近期取得的巨大进展。无论基于非线性还是线性代数稳定化方法，目前仍是唯一能在对流主导演化方程中同时实现全局DMP与高精度数值结果的有限元方法。