International Workshop on Numerical Analysis of Singularly Perturbed Differential Equations

Tufts University, Medford MA

Investigators

Abstract

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.