Mathematical and Numerical Analysis of Asymptotically Compatible Discretization of Nonlocal Models

非局部模型的研究在材料科学、力学、生物学和社会科学等许多科学和工程学科中引起了极大的关注，因此也是应用数学和计算数学的研究热点。非局部模型不同于更常见的局部模型，因为它们考虑了某一范围内的活跃因素，而不仅仅是考虑某一点上的因素。该项目的目的是促进对有限范围内相互作用的非局部模型进行稳健和有效的数值方法的数学和数值分析。这项研究将补充正在开发的有效的仿真平台，用于各种应用领域的非局部建模。这也将有助于学生的跨学科综合教育和研究培训。渐近相容(AC)离散格式提供了一类重要的非局部模型的稳健数值格式。后者的设计是为了确保近似解的收敛，随着数值分辨率的提高，以校正具有变化甚至减小的非局部相互作用范围的问题的物理解。该项目将包括对非局部问题的AC方案的全面研究，该问题具有非局部相互作用的非均匀分布范围和/或具有边界/界面。对于涉及局部/非局部耦合模型的问题和由重要应用引起的非线性问题，将进一步研究交流格式。对像AC方案这样的稳健离散化方法的关注与可靠和高效的非局部模型的模拟尤其相关，这些模型应用于涉及多尺度、奇异和反常行为的复杂物理系统。综合分析和计算方法将被用来发展基本的想法和实践的洞察力，因此研究结果不仅将丰富空调方案的数学理论，还将为其实际应用提供指导。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The study of nonlocal models has attracted much attention in many science and engineering disciplines such as materials science, mechanics, biology, and social science, and they are therefore of interest to applied and computational mathematics. Nonlocal models differ from the more common local models because they account for the factors active on a range rather than only at a point at which they are considered. The project is aimed at advancing the mathematical and numerical analysis of robust and effective numerical methods for those nonlocal models with a finite range of interactions. The research will complement the ongoing development of effective simulation platforms for nonlocal modeling in various application domains. It will also contribute to the integrated interdisciplinary education and research training of students.An important class of robust numerical schemes for nonlocal models is provided by asymptotically compatible (AC) discretization schemes. The latter are designed to assure the convergence of approximate solutions, as numerical resolution gets refined, to correct physical solutions for problems with changing or even diminishing ranges of nonlocal interactions. The project will include a comprehensive study of AC schemes for nonlocal problems with heterogeneously distributed ranges of nonlocal interactions and/or having boundary/interfaces. Further investigations of AC schemes will be carried out for problems involving coupled local/nonlocal models and nonlinear problems motivated by important applications. The focus on robust discretization methods like the AC schemes is particularly relevant to reliable and efficient simulations of nonlocal models with application to complex physical systems involving multiscale, singular, and anomalous behaviors. An integrated analytical and computational approach will be used to develop both fundamental ideas and practical insight so that the research findings will not only enrich the mathematical theory of AC schemes but also offer guidance to their practical applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.