SIAM Review, Volume 66, Issue 1, Page 1-1, February 2024.
Numerical methods for partial differential equations can only be successful if their numerical solutions reflect fundamental properties of the physical solution of the respective PDE. For convection-diffusion equations, the conservation of some specific scalar quantities is crucial. When physical solutions satisfy maximum principles representing physical bounds, then the numerical solutions should respect the same bounds. In a mathematical setting, this requirement is known as the discrete maximum principle (DMP). Discretizations which fail to fulfill the DMP are prone to numerical solutions with unphysical values, e.g., spurious oscillations. However, when convection largely dominates diffusion, many discretization methods do not satisfy a DMP. In the only article of the Survey and Review section of this issue, “Finite Element Methods Respecting the Discrete Maximum Principle for Convection-Diffusion Equations,” Gabriel R. Barrenechea, Volker John, and Petr Knobloch study and analyze finite element methods that succeed in complying with DMP while providing accurate numerical solutions at the same time. This is a nontrivial task and, thus, even for the steady-state problem there are only a few such discretizations, all of them nonlinear. Most of these methods have been developed quite recently, so that the presentation highlights the state of the art and spotlights the huge progress accomplished in recent years. The goal of the paper consists in providing a survey on finite element methods that satisfy local or global DMPs for linear elliptic or parabolic problems. It is worth reading for a large audience.

中文翻译：

---

调查与回顾

SIAM Review，第 66 卷，第 1 期，第 1-1 页，2024 年 2 月。
偏微分方程的数值方法只有在其数值解反映了相应偏微分方程物理解的基本属性时才能成功。对于对流扩散方程，一些特定标量的守恒至关重要。当物理解满足表示物理边界的最大原则时，数值解应该遵循相同的边界。在数学设置中，此要求称为离散极大值原理 (DMP)。无法满足 DMP 的离散化很容易出现具有非物理值的数值解，例如寄生振荡。然而，当对流在很大程度上主导扩散时，许多离散化方法不满足 DMP。在本期调查与评论部分的唯一一篇文章“尊重对流扩散方程离散最大原理的有限元方法”中，Gabriel R. Barrenechea、Volker John 和 Petr Knobloch 研究并分析了在符合DMP的同时提供精确的数值解。这是一项不平凡的任务，因此，即使对于稳态问题，也只有少数这样的离散化，而且都是非线性的。这些方法大多数是最近才开发出来的，因此本次演示强调了最新技术并强调了近年来取得的巨大进步。本文的目标在于提供对满足线性椭圆或抛物线问题的局部或全局 DMP 的有限元方法的调查。值得广大读者阅读。