微分复形与数值稳定性
Differential complexes and numerical stability
AI总结：
本文阐述微分复形在偏微分方程数值方法设计中的关键作用，强调离散微分复形对实现稳定离散化的重要性，并指出该几何视角为创新方法提供统一理解及攻克难题的潜力。
摘要：
微分复形（如德拉姆复形）近年来在偏微分方程数值方法的设计与分析中发挥着重要作用。偏微分方程系统的稳定离散化设计，往往取决于在离散化中捕捉系统结构的微妙方面。在许多情况下，由微分复形所捕捉的微分几何结构已被证明是一个关键要素，而与原始复形适当相关的离散微分复形至关重要。这一新的几何视角为近几十年来发展起来的各种创新数值方法提供了统一的理解，并为那些此前未知稳定离散化的问题指明了方向，而且它似乎很可能在攻克数值偏微分方程中一些当前棘手的问题方面发挥重要作用。
英文摘要：
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial differential equations often hinges on capturing subtle aspects of the structure of the system in the discretization. In many cases the differential geometric structure captured by a differential complex has proven to be a key element, and a discrete differential complex which is appropriately related to the original complex is essential. This new geometric viewpoint has provided a unifying understanding of a variety of innovative numerical methods developed over recent decades and pointed the way to stable discretizations of problems for which none were previously known, and it appears likely to play an important role in attacking some currently intractable problems in numerical PDE.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
