SIAM Review, Volume 66, Issue 1, Page 3-88, February 2024.
Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solutions 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 the 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 the 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 both 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. Similarly, methods based on algebraic stabilization, both nonlinear and linear, are currently 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 scenario.

中文翻译：

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遵守对流扩散方程离散极大值原理的有限元方法

SIAM Review，第 66 卷，第 1 期，第 3-88 页，2024 年 2 月。
对流扩散反应方程对标量守恒进行建模。从解析的角度来看，这些方程的解在某些条件下满足极大值原理，它代表了解的物理界限。在实践中，解决方案的数值近似遵循相同的界限通常是至关重要的。该属性的数学公式称为离散最大原理 (DMP)，它有助于方法的物理一致性。在许多应用中，对流主导扩散几个数量级。众所周知，标准离散化通常不能满足这种对流主导状态下的 DMP。事实上，在这种情况下，构建离散化是一个具有挑战性的问题，一方面尊重 DMP，另一方面计算准确的解决方案。本文对有限元方法进行了综述，主要关注满足局部或全局 DMP 的对流主导状态。讨论了基础数值分析的概念。调查显示，对于稳态问题，只有少数离散化，而且都是非线性的，同时满足 DMP 并计算出相当准确的解，例如代数稳定方案。此外，大多数离散化都是近年来开发的，显示了最近取得的巨大进展。同样，基于代数稳定的方法（非线性和线性）是目前唯一将全局 DMP 的满足性与对流主导场景中演化方程的精确数值结果结合起来的有限元方法。