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广泛的关注，因此也引起了应用数学和计算数学的兴趣。非局部模型不同于更常见的局部模型，因为它们考虑的是在一个范围内活动的因素，而不仅仅是在它们被考虑的点上。该项目旨在推进具有有限范围相互作用的非局部模型的鲁棒和有效的数值方法的数学和数值分析。该研究将补充各种应用领域中有效的非局部建模仿真平台的持续发展。它还将有助于学生的综合跨学科教育和研究培训。渐近相容（AC）离散化为非局部模型提供了一类重要的鲁棒数值格式。后者的设计是为了保证近似解的收敛性，随着数值分辨率的提高，修正非局部相互作用范围变化甚至缩小的问题的物理解。该项目将包括对非局部问题的AC方案的全面研究，这些问题具有非局部相互作用的异构分布范围和/或具有边界/界面。交流格式将进一步研究涉及耦合局部/非局部模型的问题以及由重要应用驱动的非线性问题。像AC方案这样的鲁棒离散化方法的重点是可靠和有效地模拟非局部模型，并应用于涉及多尺度，奇异和异常行为的复杂物理系统。本文将运用分析与计算相结合的方法来发展基本思想和实践见解，使研究成果不仅丰富了交流方案的数学理论，而且对其实际应用具有指导意义。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.