[论文解读] Finite element methods respecting the discrete maximum principle for convection-diffusion equations

Gabriel R. Barrenechea, Volker John+1 more|arXiv (Cornell University)|Apr 15, 2022

Advanced Numerical Methods in Computational Mathematics被引用 6

一句话总结

本文综述了在对流主导区域中满足离散最大值原理（DMP）的有限元方法，针对对流-扩散-反应方程。它指出，代数稳定化方案以及像FEM-FCT和AFC这样的非线性方法是目前唯一能同时保持DMP并提供精确解的方法，突出了单调性高阶离散化方面的最新进展。

ABSTRACT

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.

研究动机与目标

• 识别并分析在对流主导的对流-扩散问题中严格满足离散最大值原理（DMP）的有限元方法。
• 评估稳定化有限元格式中DMP满足性与数值精度之间的权衡。
• 评估线性与非线性稳定化技术在保持解的物理边界方面的性能。
• 总结代数稳定化和通量校正有限元方法在稳态和时变问题中的最新进展。
• 全面概述确保解的保正性和避免对流主导流中虚假振荡的方法。

提出的方法

• 聚焦于通过代数稳定化（包括人工扩散和局部投影技术）强制实施DMP的有限元方法。
• 分析稳态问题中的非线性格式，如Mizukami–Hughes方法、Burman–Ern方法以及代数通量校正（AFC）方法。
• 研究时变问题中的FEM-FCT和单调局部投影稳定化（LPS）方法，强调其保持DMP的能力。
• 回顾线性方法，如迎风法和边平均有限元法，指出其在对流主导条件下难以满足DMP的局限性。
• 评估限制器和梯度近似在提升精度的同时保持单调性和线性保持性的作用。
• 比较不同有限元类型（包括$\mathbb{Q}_1$、高阶$H^1$-协调、Crouzeix–Raviart以及不连续伽辽金元）的方法性能。

实验结果

研究问题

• RQ1 在对流主导区域中，哪些对流-扩散方程的有限元方法能够严格满足离散最大值原理（DMP）？
• RQ2 与经典稳定化方法相比，代数稳定化方案在DMP满足性和解精度方面表现如何？
• RQ3 像通量校正和局部投影这样的非线性稳定化机制在保持单调性和避免振荡方面起什么作用？
• RQ4 线性有限元方法能否被修改以在保持高阶精度的同时满足全局DMP？
• RQ5 在三维中实现DMP保持方法的实际权衡是什么，特别是计算成本和非线性求解器效率方面？

主要发现

• 在稳态方法中，只有非线性格式——特别是代数稳定化和通量校正方法——能够同时满足DMP并提供精确解。
• Mizukami–Hughes方法是唯一被证明能保持DMP的经典稳定化方法，而大多数其他方法在对流主导条件下会失效。
• 近期的代数稳定化方案，包括改进的LPS和基于梯度的限制器，显示出更高的精度，且不会过度弥散，同时减少了阶梯效应。
• 对于时变问题，代数稳定化FEM-FCT和AFC格式是目前唯一能保持全局DMP并维持良好精度的有限元方法。
• 代数稳定化的线性变体在精度和计算效率之间提供了有利的平衡，使其适用于实际应用。
• 在半离散格式中引入单调局部投影算子可显著减少阶梯效应并提升解的质量，同时不损害DMP的保持性。