International Workshop on Numerical Analysis of Singularly Perturbed Differential Equations

一般来说，计算数学，特别是数值分析和科学计算（NASC），现在被公认为提供了科学研究的“第三种模式”，补充了传统的理论和实验方法。虽然NASC在基础科学和工业上都取得了许多成功，但一类仍然具有挑战性的问题是那些模拟量快速变化的问题，例如流体动力学和其他学科中出现的边界和内层。该基金支持美国参加于2016年7月25日至29日在加拿大哈利法克斯圣玛丽大学校园举行的“奇摄动微分方程数值分析”国际研讨会和短期课程。NSF的资金将允许美国的参与者从学习该领域的基本工具和最先进的技术中受益，并与来自加拿大和国外的研究人员开展新的合作。PI将优先支持来自代表性不足群体的参与者。边界层和内层的问题出现在科学和工业模拟的许多领域，包括数学生物学、流体动力学和电磁学。这些领域需要有效的参数鲁棒仿真工具来支持科学家和工程师感兴趣的应用问题日益增加的复杂性。从广义上讲，这些问题可以被归类为“奇异摄动问题”，其中确定了一个小参数来控制这些临界层的宽度。当这些参数趋近于零时，几乎所有的标准离散化和网格细化策略都不能产生鲁棒误差控制；因此，NASC的一个专门的子领域已经出现，以解决奇摄动微分方程的有效和鲁棒模拟。这个为期两天的短期课程将专注于拟合层网格的有限差分和有限元离散化，以及针对这些问题的自适应网格细化策略。讲习班的其余部分旨在在讲习班参与者之间进行合作研究会议，以将这些工具应用于现实世界的问题。拟议的短期课程和研讨会有许多机会产生更广泛的影响，包括发展新的跨学科研究联系。

Computational Mathematics, in general, and Numerical Analysis and Scientific Computing (NASC), in particular, are now well-recognized as providing a "third mode" of scientific investigation, complementing traditional theoretical and experimental approaches. While the successes of NASC in both basic science and industry are many, one class of problems that remains challenging are those with rapid changes in the simulated quantities, such as arise with boundary and interior layers in fluid dynamics and other disciplines. This grant supports US participation in an international workshop and short course on "Numerical Analysis for Singularly Perturbed Differential Equations", to be held on the campus of St. Mary's University in Halifax (Canada), from July 25-29, 2016. NSF funds will allow US-based participants to benefit from learning both the basic tools in this area and the state-of-the-art, developing new collaborations with researchers from Canada and abroad. The PI will prioritize support for participants from underrepresented groups.Problems with boundary and interior layers arise in many areas of scientific and industrial simulation, including mathematical biology, fluid dynamics, and electromagnetics. Efficient parameter-robust simulation tools are needed in these fields to support the ever-increasing complexity of applied problems of interest to scientists and engineers. Broadly, these problems can be categorized as ``singular perturbation problems", where a small parameter is identified that controls the width of these critical layers. In the limit as these parameters approach zero, nearly all standard discretization and mesh-refinement strategies fail to yield robust error control; thus, a specialized subfield of NASC has arisen to address efficient and robust simulation of singularly perturbed differential equations. This two-day short course will focus on finite-difference and finite-element discretizations on fitted-layer meshes, as well as adaptive mesh refinement strategies for these problems. The remainder of the workshop is aimed at collaborative research sessions among workshop participants to apply these tools to real-world problems. There are many opportunities for broader impacts of the proposed short course and workshop, including the development of new interdisciplinary research connections.