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在基础科学和工业方面取得了许多成功，但仍然存在一类具有挑战性的问题，即模拟量快速变化的问题，例如流体动力学和其他学科中的边界层和内层。 该补助金支持美国参加2016年7月25日至29日在加拿大哈利法克斯玛丽大学校园举行的国际研讨会和“奇异摄动微分方程数值分析”短期课程。 NSF的资金将使美国的参与者受益于学习这一领域的基本工具和最先进的技术，并与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的群体的参与者。边界层和内部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。 在这些领域需要有效的参数鲁棒仿真工具，以支持科学家和工程师感兴趣的应用问题的日益复杂性。 广义地说，这些问题可以归类为“奇异摄动问题”，其中确定了控制这些临界层宽度的小参数。 在这些参数接近于零的极限，几乎所有的标准离散化和网格细化策略都不能产生鲁棒的误差控制，因此，NASC的一个专门的子领域已经出现，以解决有效和鲁棒的模拟奇摄动微分方程。这个为期两天的短期课程将集中在有限差分和有限元离散化的拟合层网格，以及自适应网格细化策略，这些问题。研讨会的其余部分旨在研讨会参与者之间的合作研究会议，将这些工具应用于现实世界的问题。 拟议的短期课程和讲习班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.