Survey and Review

Abstract

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.