关于满足比较原理、最大原理和非负约束的网格限制：近期进展和新结果
On mesh restrictions to satisfy comparison principles, maximum principles, and the non-negative constraint: Recent developments and new results
AI总结：
本文研究了为一般线性二阶椭圆偏微分方程满足最大原理、比较原理和非负约束所需的网格限制，推导了新结果并讨论了未来研究方向。
摘要：
本文探讨了为一般线性二阶椭圆偏微分方程满足最大原理、比较原理和非负约束所需的网格限制。我们回顾了该领域最近的发展，推导了新结果，并讨论了未来的研究方向。特别地，我们推导了三节点三角形（T3）元素和四节点四边形（Q4）元素在标准单场伽辽金格式下满足比较原理、最大原理和非负约束的限制。分析仅限于均匀椭圆线性微分算子（散度形式）和整个域边界上的狄利克雷边界条件。讨论了连续和离散设置中的各种最大原理和比较原理版本。文献中已知，锐角三角形足以满足纯各向同性扩散的离散弱最大原理。开发了迭代算法，用于构造保持离散最大原理的单纯形网格，利用现有的开源网格生成器。展示了不同三角化类型的数值例子，以说明对计算网格施加限制的优缺点。还通过代表性数值例子量化了局部和全局质量守恒误差，并展示了基于度量的网格在质量守恒方面的性能。
英文摘要：
This paper concerns with mesh restrictions that are needed to satisfy several important mathematical properties -- maximum principles, comparison principles, and the non-negative constraint -- for a general linear second-order elliptic partial differential equation. We critically review some recent developments in the field of discrete maximum principles, derive new results, and discuss some possible future research directions in this area. In particular, we derive restrictions for a three-node triangular (T3) element and a four-node quadrilateral (Q4) element to satisfy comparison principles, maximum principles, and the non-negative constraint under the standard single-field Galerkin formulation. Analysis is restricted to uniformly elliptic linear differential operators in divergence form with Dirichlet boundary conditions specified on the entire boundary of the domain. Various versions of maximum principles and comparison principles are discussed in both continuous and discrete settings. In the literature, it is well-known that an acute-angled triangle is sufficient to satisfy the discrete weak maximum principle for pure isotropic diffusion. An iterative algorithm is developed to construct simplicial meshes that preserves discrete maximum principles using existing open source mesh generators. Various numerical examples based on different types of triangulations are presented to show the pros and cons of placing restrictions on a computational mesh. We also quantify local and global mass conservation errors using representative numerical examples, and illustrate the performance of metric-based meshes with respect to mass conservation.
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